UFDC Home myUFDC Home  |   Help
<%BANNER%>

# Asymptotic Enumeration in Pattern Avoidance and in the Theory of Set Partitions and Asymptotic Uniformity

## Material Information

Title: Asymptotic Enumeration in Pattern Avoidance and in the Theory of Set Partitions and Asymptotic Uniformity
Physical Description: 1 online resource (91 p.)
Language: english
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

## Subjects

Subjects / Keywords: asymptotic, avoidance, bell, distribution, normal, packing, pattern, permutation, stirling, uniform
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

## Notes

Abstract: We demonstrate asymptotic properties of some popular combinatorial objects, including a partial answer to an open problem posed by Michael Atkinson and a general result on conditions for the coincidence of asymptotic normality and uniformity. We say the permutation pi contains the pattern sigma if there exists a subsequence of pi order isomorphic to sigma. Denote by s_n(tau, sigma) the number of permutations of length n which do not contain either of the patterns sigma and tau. For a pattern sigma of length m, we denote by sigma' the pattern (m+1)sigma and construct classes of patterns for which the limit supremum of s_n(123 ... r,sigma)^(1/n) agrees with the limit supremum of s_n(123 ... r,sigma')^(1/n) for several classes of patterns sigma. We also construct classes of permutations which avoid 123 ... r and contain 'many' patterns. Many combinatorial sequences are of the form (a_{n,k}) where n ranges over the non-negative integers and, for each n, there exists m = m(n) such that k ranges from 1 to m. We call such a sequence a combinatorial distribution. Many combinatorial distributions, upon rescaling, approach in distribution the normal distribution as \$n\$ grows to infinity, a phenomenon we call asymptotic normality. A combinatorial distribution is said to be asymptotically uniform if, for each positive integer q and each residue class modulo q, the sum of coefficients a_{n,k} with k congruent to r (mod q) approaches 1/q as n grows to infinity. We call this asymptotic uniformity. We prove that if the generating polynomials for a combinatorial distribution have real, nonnegative zeros, asymptotic normality implies asymptotic uniformity. We apply this result to several sequences from the literature. Finally, we present original results on the zeros of the Bell polynomials which were first attained in proving the asymptotic uniformity of the Stirling numbers of the second kind.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Bona, Miklos.

## Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022066:00001

## Material Information

Title: Asymptotic Enumeration in Pattern Avoidance and in the Theory of Set Partitions and Asymptotic Uniformity
Physical Description: 1 online resource (91 p.)
Language: english
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

## Subjects

Subjects / Keywords: asymptotic, avoidance, bell, distribution, normal, packing, pattern, permutation, stirling, uniform
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

## Notes

Abstract: We demonstrate asymptotic properties of some popular combinatorial objects, including a partial answer to an open problem posed by Michael Atkinson and a general result on conditions for the coincidence of asymptotic normality and uniformity. We say the permutation pi contains the pattern sigma if there exists a subsequence of pi order isomorphic to sigma. Denote by s_n(tau, sigma) the number of permutations of length n which do not contain either of the patterns sigma and tau. For a pattern sigma of length m, we denote by sigma' the pattern (m+1)sigma and construct classes of patterns for which the limit supremum of s_n(123 ... r,sigma)^(1/n) agrees with the limit supremum of s_n(123 ... r,sigma')^(1/n) for several classes of patterns sigma. We also construct classes of permutations which avoid 123 ... r and contain 'many' patterns. Many combinatorial sequences are of the form (a_{n,k}) where n ranges over the non-negative integers and, for each n, there exists m = m(n) such that k ranges from 1 to m. We call such a sequence a combinatorial distribution. Many combinatorial distributions, upon rescaling, approach in distribution the normal distribution as \$n\$ grows to infinity, a phenomenon we call asymptotic normality. A combinatorial distribution is said to be asymptotically uniform if, for each positive integer q and each residue class modulo q, the sum of coefficients a_{n,k} with k congruent to r (mod q) approaches 1/q as n grows to infinity. We call this asymptotic uniformity. We prove that if the generating polynomials for a combinatorial distribution have real, nonnegative zeros, asymptotic normality implies asymptotic uniformity. We apply this result to several sequences from the literature. Finally, we present original results on the zeros of the Bell polynomials which were first attained in proving the asymptotic uniformity of the Stirling numbers of the second kind.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Bona, Miklos.

## Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022066:00001

## This item has the following downloads:

Full Text

ASYMPTOTIC ENUMERATION IN PATTERN AVOIDANCE AND IN THE THEORY
OF SET PARTITIONS AND ASYMPTOTIC UNIFORMITY

By

MICAH SPENCER COLEMAN

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2008

S2008 Micah Spencer Coleman

I dedicate this dissertation with love and pride to the memory of Juliana Cole, whose

curiosity and lifelong devotion to learning have defined my family and instilled in me what

was critical to survive my graduate career. Enjoy the '., -_ipes, Grandmother.

ACKNOWLEDGMENTS

My deepest love and admiration to my wife Hiroko. While studying in a foreign

language in a foreign land, she took up two of the hardest imaginable roles, that of

military spouse and that of mathematician caretaker. Daisuki! I thank my parents Bob

and Bobbi Coleman, my brother Matt, and mi vecino Abby for their patience, humor, and

integrity. Great thanks go to Professors Julie Miller, Tina Carter, and Norm Levin, for

first introducing me to I ii iii to our Graduate Coordinator Paul Robinson, and to

the greatest advisory committee ever assembled, Professors David Drake, Meera Sitharam,

Andrew Vince, and Neil White. I am honored and humbled to be associated with each of

them. Finally, my deepest respect and appreciation are held for my advisor, B6na Mikl6s.

page

ACKNOW LEDGMENTS ................................. 4

LIST OF FIGURES .................................... 7

A B ST R A CT . . . . . . . . .. . 8

CHAPTER

1 INTRODUCTION ...................... .......... 10

1.1 Asymptotic Enumeration ............................ 10
1.2 Notation for Asymptotic Growth Rates ......... ........... 10
1.3 Generating Functions .................. ......... .. .. 11

2 PATTERN AVOIDANCE IN PERMUTATIONS AVOIDING A MONOTONE
PATTERN . . . . . . . . .. 12

2.1 Permutations and Permutation Patterns ........ ........... 12
2.2 An Open Problem by M. Atkinson .................. ..... 24
2.3 Generating Trees .................. ............. .. 26
2.4 "Hat" Notation .................. .............. .. 31
2.5 Monotone Increasing Patterns q ........ ........ .. .. 33
2.6 The Pattern q = 123 ............. .......... .. 38

3 PATTERN PACKING ........... ..... . ..... .. 53

3.1 General Pattern Packing .......... . . ... 53
3.2 Pattern Packing in 123-avoiding Permutations . . ..... 57
3.3 Pattern Packing in q-avoiding Permutations ................ .. 59
3.4 Packing Density and Further Directions .............. .. .. 61

4 ASYMPTOTIC NORMALITY AND UNIFORMITY . . ..... 63

4.1 Probability Theory .................. ............ .. 63
4.2 Triangular Arrays .................. ............. .. 64
4.3 Asymptotic Normality .................. .......... .. 65
4.4 Asymptotic Uniformity ..... . . ..... ........... 66
4.5 Generating Polynomials with Real, Non-Positive Roots . .... 68
4.6 Asymptotic Normality Implies Asymptotic Uniformity . ... 71

5 ON THE ROOTS OF THE BELL POLYNOMIALS . . ..... 74

5.1 Stirling Numbers of the Second Kind ................ .. .. 74
5.2 Bell Polynomials ............... . . .... 77
5.3 Bounds on the Roots of the Bell Polynomials . . ...... 80
5.4 Asymptotics of the Roots of the Bell Polynomials . . ..... 83

REFERENCES ....................................... 88

BIOGRAPHICAL SKETCH .......... ... ................ 91

LIST OF FIGURES

Figure

2-1 The permutation 3142. .........

2-2 The permutation 532614 . .

2-3 The permutation 865321947 . .

2-4 A rooted tree. ............

2-5 The complete binary tree . .

2-6 The Fibonacci tree . .....

2-7 T(123,132). .............

2-8 Tree in W(123,231) rooted at 42153 .

2-9 The 1 it-, i permutation 213654 with

3-1 The permutation W(6) = 342516 .

lv, i-rs 21, 3,

and 654

page

13

25

27

28

29

30

32

40

41

55

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

ASYMPTOTIC ENUMERATION IN PATTERN AVOIDANCE AND IN THE THEORY
OF SET PARTITIONS AND ASYMPTOTIC UNIFORMITY

By

Micah Spencer Coleman

May 2008

('C! r: Mikl6s B6na
Major: Mathematics

We demonstrate ..-i-! iiill ic properties of some popular combinatorial objects,

including a partial answer to an open problem posed by Michael Atkinson and a general

result on conditions for the coincidence of ..i-mptotic normality and uniformity.

For a permutation 7r S, written in one line notation as 7r = 7Tr2 Tr, we ,-i

7 contains the pattern a E Sm if there exists a subsequence 7i, such that for all

1 < j, k < m it holds that Ti, < 7 if and only if ca < Uk. Denote by s,((, a) the number

of n-permutations which do not contain either of the patterns a and r. Letting a' denote

the pattern (m + 1)a, we construct classes of patterns for which the limit supremum of

sT(123 .* r, a) /" agrees with the limit supremum of sT(123 .. r, a')1/" for several classes

of patterns a. We also construct classes of permutations which avoid 123 ... r and contain

i,,r, '': patterns.

iM ,I: combinatorial sequences are of the form (aT,k) where n ranges over the

non-negative integers N and, for each n, there exists m = m(n) such that k ranges

from 1 to m. We call such a sequence a combinatorial distribution. M ilv combinatorial

distributions, upon rescaling, approach in distribution the normal distribution as n grows

to infinity, a phenomenon we call '-;,,,/'l/. '.: no I ,,,l.:/, A combinatorial distribution is

said to be i~-;,,'/,/I J.. all ;, u [.,rm if, for each positive integer q and each residue class

modulo q, the sum of coefficients an,k with k r (mod q) approaches 1/q as n grows to

infinity. We call this 1'- ;,/,/./ : unifoiiii;, We prove that if the generating polynomials

for a combinatorial distribution have real, nonnegative zeros, .,-i-~ii l, I ic normality implies

.-i-!,ill I ic uniformity. We apply this result to several sequences from the literature.

Finally, we present original results on the zeros of the Bell polynomials which were

first attained in proving the .- i:!! I,.1 ic uniformity of the Stirling numbers of the second

kind Sn,k.

CHAPTER 1
INTRODUCTION

1.1 Asymptotic Enumeration

Enumerative combinatorics involves counting discrete objects, determining the

cardinalities of sets which are indexed by one or more integers. In many cases exact

formulae are known. However, some formulae are so convoluted as to obscure their most

telling information. In either of these cases .. -mptotics provide powerful methods for

understanding the classes under question.

With that said, we could define .,-iii!,l ,l ic combinatorics as the study of the growth

of combinatorial sequences indexed by an integer n as n grows without bound, n -- oo.

For a survey of these techniques, the reader is reffered to the chapter by Odlyzko [26] in

the Handbook of Combinatorics.

Definition 1. We introduce some notation to be used throughout. Let [n] denote the set

{1, 2,... n}, and [a, b] denotes the set {a, a + 1,..., b}. N denotes the natural numbers

n > 0, while P denotes the positive integers n > 1.

1.2 Notation for Asymptotic Growth Rates

Let f : P P be a function with some sort of predictable behavior. What does it

mean to understand the ..I-:ii-!1.1 ic behavior or growth of f? If it is anything like most

functions we encounter, as n grows arbitrarily large, the growth of f probably follows some

pattern, for example a straight line or logarithmic curve. Perhaps f acts erratically on

every small interval but has a smooth overall growth which we can mimic with some other

function g which is easier to understand.

Example 1. D. I,,- the function ((n) = n2 + (-1). This function is ..-, to write but

not so ..r-; to ig'''l, However, as n grows very I7",,, we see that ((n) is very close to the
'"* 'lJi'. 'ii !, Jl n 2.

Such a situation motivates Big 0 notation. Given two functions f and g defined on

the positive integers, we write

f(n) = O(g(n))

if there exists a nonzero constant M such that If(n)| < IMg(n)| for all n > 1.

Similarly,

f(n) = (g(n))

if there exists a constant M such that If(n)l > IMg(n) for all n > 1.

Finally,

f (n) ~ (n)

if
lim f(n) 1
n-oo g(n)
holds.

1.3 Generating Functions

A powerful area of combinatorics is Generatingfunctionology. For surveys of the field,

the reader is referred to the texts by Wilf [40], Stanley [33], [34], and B6na [8].

Definition 2. Given a sequence (an)n>o, the associated ordinary generating function is

the power series

f(x) = anxT.
n>O
Similarly, for a sequence (a,,k) 1, ,.,' l for all n > 0 and 0 < k < m(n) for some function

m(n), the associated generating polynomial for each n is the I" l;'l, ,.;;'.:l
m(n)
,q(x) Z Y ank.
k>O

The power is in handling these power series and polynomials to reveal information

about the sequence in question. Generating polynomials will p1 iv a fundamental role in

the last two chapters.

CHAPTER 2
PATTERN AVOIDANCE IN PERMUTATIONS AVOIDING A MONOTONE PATTERN

2.1 Permutations and Permutation Patterns

One aspect of enumerative combinatorics, one of the simple beauties that make

it such an attractive discipline, is the ease with which it can be explained to the

non-specialist or even non-mathematician / non-scientist. We seek to count the objects in

some class of size n. For example, we may estimate the number of objects in some class

which are of size n and have statistics 1, k as n grows arbitrarily large. In fact, there are

some combinatorial objects which are more easily explained to a non-mathematician than

to some of our mathematical colleagues. The case in mind is the permutation. In the

field of permutation patterns, one views a permutation in one-line notation; i.e., as an

arrangement of the numbers 1 through n for some n. That is it. While we readily admire

and appreciate the wisdom and complexity of our friends the algebraists and topologists,

there is a certain frustration and loss of momentum when trying to describe the simple

properties of permutations that we are dealing with here to such an audience. There is no

consideration of cycle structure, what set is acted upon, etc. From such a rough definition

we can pose many of the fundamental questions dealt with in this field of research.

In the first three chapters of this dissertation we introduce the concept of a

permutation pattern, survey some of the 1n i, r results and trends in the field of

permutation patterns, develop a foundation for the work contained herein, and pose

further questions to be explored.

Definition 3 (rough). A permutation of length n or a permutation on [n] or an

n-permutation is an arr i.,, i,, n,: of the numbers 1 through n.

Definition 4 (rigorous). An n-permutation r is an injective I'""Ij'.:'" from the set

[n] = 1, 2,..., n} onto itself. We denote 7(i) by it and write w as the concatenation

7T172 7' IT l .':/ the numbers 71 ,... ,7r the entries of r. It should be noted that by

convention we allow there to be one permutation on the mpl"; set. F.:,.11, for each n > 0

we denote by S, the set of n-permutations.

We use the convention that an ascent in a permutation 7~ is the index of an entry 7~

with 7i < 7i+,. Likewise a descent in 7~ is the index of an entry 7ri with 7ri > 7i+1. We

define an ascendee (resp. descended) to be the index of an entry 7~ with 7~i- < T (resp.

7i-1 > ii).

Example 2. For ~ = 3142, 2 is an ascent; 1 and 3 are descents. 3 is an ascendee; 2 and

4 are descendees. As these are geometric as opposed to ri/, braic concepts, they i,,~ni best be

understood ;.',all;i as in Fr.:';,,, 2-1
4

3

2

1

Figure 2-1. The permutation 3142.

In all contexts considered here, a permutation pattern or simply a pattern is itself a

permutation but there are subtle differences which we shall exploit. Given a permutation

7T = 1 TT,, a subsequence of 7T is an ordered subset of the entries of T, ( T~T ,..., ) for

some k, which we write in the same order as they appear in r, so il < i2 < < ik.

Example 3. Consider the 5-permutation r = 13425. If we want to construct a subse-

quence, we have two choices of what to do with each entry 7i, whether to include 7r in our

subsequence or not. Therefore, we have 25 = 32 subsequences of 7. We list them here in an

obvious but i, i,'.',.l. -l. ordering:

Length 0 : 0

Length 1: 1 3 4 25

Length 2 13 14 12 15 34 32 35 42 45 25

Length 3: 134 132 135 142 145 125 342 345 325 425

Length 4 1342 1345 1325 1425 3425

Length 5 : 13425

Roughly speaking, each such subsequence can be associated with some pattern, and

from this we generate vast fields of research. With such I-Jlu: i let us formalize the

concept of the permutation pattern.

Definition 5. We call a finite sequence X of distinct positive integers reduced if the

elements of the sequence form the set [n] for some n, i.e. if X is a permutation of [n].

Let X = X1X2 ... X, be a finite sequence of n distinct positive integers. Then,

there exists a unique permutation 7r E S, such that for all pairs of indices i and j with

1 < i
Xi < Xj if and only if 7ri < rj.

So, X and 7r are in the same "order", or order isomorphic. As we assumed 7r to be a

permutation in S,, 7r is reduced, and we call 7r the reduction of X.

Definition 6. Let 7 E S,. We -.,;/ that 7 contains the pattern a E Sm (m < n) if there

exists a subsequence 7 i- .. 7 (i < i2 < ... < i,) whose reduction is a. If there is no

such subsequence, we -.,;, that 7 avoids the pattern a.

One more note is in order on the above definition, which implicitly defines patterns.

A pattern is itself simply a permutation, but we use the two words to describe distinct

sets. We ask questions such as does the permutation 7 contain the pattern a, etc. These

meanings will be clear in the context. Please also note that we will use the convention

wherever possible of letting m denote the length of a pattern a and letting n denote the

length of a permutation 7 for which we want to determine a-avoidance, etc. Much of

this dissertation will be devoted to questions surrounding the avoidance of a set of two

patterns, one of which will be monotone increasing. We will conventionally let q denote

the monotone increasing pattern in question and let r denote its length.

Example 4. The permutation 34521 contains the pattern 123, as the first three entries of

34521 are 345, which reduces to 123.

Example 5. We exhaust all patterns contained in the permutation 13425 from Example 3

by reducing all subsequences, in the same ordering as above:

Length 0 : 0

Length 1:1 1 1 1 1

Length 2: 12 12 12 12 12 21 12 21 12 12

Length 3: 123 132 123 132 123 123 231 123 213 213

Length 4: 1342 1234 1324 1324 2314

Length 5 : 13425

Of course, we could continue in this fashion. It is a great exercise for the beginner

in this area to exhaust the subsequences of a permutation to determine what patterns

the permutation contains or avoids and pose some conjectures. This is how one learns

anything in combinatorics, by w 11 ig our hands dirty", doing enough manual labor on

our combinatorial objects to get a feel for their growth and other properties.

Definition 7. Let a E S, be a pattern. For each n > 0 we denote by S,(cr) the set

of all n-permutations which avoid a, and write s,(a) = S, (a)l, i.e. the number of

n-permutations which avoid a.

More generill let E be a set of patterns. For each n > 0 we denote by S,(E)

the set of all n-permutations which avoid all patterns in E, enumerated by s,,(). If

Z = {uli 2, 2. Jk we write S,(Z) as S.(ui, 02,. ok) and likewise write s,n() as

sn(ul1, 2, *, ok), dropping the brackets.

On a historical note, the S, and s, above may have been coined in honor of Simion

and Schmidt, whose 1985 paper [31] launched pattern avoidance and contained some

results which are still hallmarks of the field.

As the name would imply, enumerative combinatorialists most enjoy enumerating

sets, that is, determining a precise formula for the cardinality of each set which depends

only on the index or indices of that set. Unfortunately, we often find quite interesting

discrete objects whose nature is complex enough to elude precise formulae. Alas, we will

see that for most patterns a, the sequence s,(a) falls into the latter category. However,

all is not lost. As was briefly discussed in the introductory chapter, great information

can still be had by the imptotics of a sequence, and many of the current results in the

field of permutation patterns involve bounds and limits which are not as strong as precise

formulae, but carry power and beauty of their own.

Let us first see some examples of patterns for which we can give an exact formula. We

will treat all patterns in S, for 0 < m < 3. We will make use of the Kronecker delta 6ij,

defined by

1 if i = j
i,j 0 if
0 ifi e/.

Proposition 6. We have the following exact formulae for a E Sm, m = 0, 1,2:

n(0) = 0,

Sn(l) = 6,o0,

s,(12) 1,

s,(21) 1.

Proof. As we can take an empty subsequence of any permutation, including the empty

permutation itself, no permutation can avoid the pattern 0, and s,(0) = 0.

Similarly, every non- n/l' ;, permutation has a non-empty subsequence of length

1, and s,(l) = 0 unless n = 0, in which case we have so(l) = 1, counting the sole

permutation in So.

Note that a permutation 7r S,, avoiding the pattern 12 is equivalent to 7 having

no ascents, implying 7 is the unique monotone descending sequence of length n, so

s,(12) = 1. The dual statement is that avoiding the pattern 21 is equivalent to avoiding

descents, thus s,(21)= 1. E

The reader may (should) have been amazed by the fact that s,(12) = s,(21) and their

dual p"... '[ In fact, the duality involved was that 12 and 21 are reverses of each other,

and for each n the unique permutation in S,(12), the monotone decreasing permutation,

is the reverse of the monotone increasing permutation, the unique element of S,(21). Of

course, one could also prove the equality with the fact that 12 and 21 are complements of

each other. Perhaps these statements also apply to longer, more interesting patterns?

Definition 8. Let 7 = 71rT2 ... 7 be a permutation. We /. I;,: the reverse, complement,

and algebraic inverse, resp., of 7 as

R
7T R= 7nn-1 7T1,

7T C (n 71l + l)(n 7"2 + 1) (n 7{n + 1),

-1 1 ... n,

where, for all 1 < i < n, r, = i.

Example 7. Let r = 24315. Then, we have the eight related permutations

r = 24315,

R = 51342,

c = 42351,

-1 41325,

7rR( 7CR) 15324,

(7R)-1 25341,

(rC)-1 52314,

(7RC) -1 14352.

Of course, the algebraic inverse r-1 is just what the name would imply, the inverse

of r considered as an element of the p'. '"' S,. Such a definition as the one above may

seem crude, but we had promised to only consider these permutations as linear orders, not

group elements.

In our example of r = 24315, we made easy use of the obvious fact that R and

C commute as operators. We leave it as an exercise that they each commute with the

algebraic inverse as well.

The definition of reverse should be clear. We note that we can write each entry of rR

as r R = 7 -i+1. We can think of complement as "flippiin the permutation in a vertical

sense. Now that we have these definitions, we return to the question of equivalences

among patterns.

Suppose a permutation 7 = 71r r, contains the pattern a = a1 ... r,. Then, we

have a set of indices 1 < pl < pm < n such that the reduction of the subsequence

pU, -7, is precisely a, i.e. for all i < j,

., < 7rp, if and only if ai < Kj.

By the remarks following Definition 8, this is equivalent to the statement that, for each

pair < j,
R R R R
-R, +1 > -I,,+1 if and only if ani+1 > a7_i,.

As we run through all pairs i < j, we see that this collection of statements is equivalent to

R > -R if and only if ae > a1.

Altogether, 7 containing a is equivalent to rR containing aR. Likewise, r containing a

is equivalent to rC containing aC and 7-1 containing a-1. Of course, we can replace the

word containingg by the word ,vpli..[Ii; in each statement.

Critically, as the three maps -R, .c and --1 are bijections S, -- Sn, it follows that

the number of n-permutations which avoid a is the same as the number of n-permutations

avoiding aR, aC, a -. We have achieved the following result.

Lemma 1. Let a be a pattern. Then, for all n > 0,

s,(ao) = s,(aR) = s,(oaC) = s,(1-~).

In fact, we can ?- that each a E Sm belongs to an equivalence class of m-patterns

whose sequences sT are equal.

Definition 9. We -.r;, the patterns a and r are Wilf-equivalent if s,(ac) = Sn,() holds for

all n > 0. This leads nature /ll/; to the 1/ fl ,.:/.:,In of the Wilf-equivalence class of a pattern a

as the set of all patterns which are Wilf-equivalent to a.

For our first non-trivial results, we move to m = 3. S3 consists of the six patterns

123, 132, 213, 231, 312, and 321. One quickly recognizes that there are at most two Wilf

equivalence classes in S3 by noting

123 = 321,

132R 231,

132 312,

132Rc 213.

In fact, there is only one Wilf equivalence class in S3 as is seen in the following

lemma.

Lemma 2. (Simion and Schmidt [31]) For all n > 0,

,(123) s(132).

Proof. We call a permutation entry which is less than all entries preceding it a left-to-right

minimum. The remaining entries we call remaining entries.

Let 7 be a 132-avoiding n-permutation. Form a new n-permutation 7' by fixing the

left-to-right minima and transposing (swapping) pairs of remaining entries until they are

in decreasing order. They will still all be remaining entries, because at each transposition

the left, smaller entry is preceded by some left-to-right minimum which will still precede

the larger remaining entry after the transposition, and the smaller remaining entry moves

to the right, still not a left-to-right minimum. Note that the left-to-right minima are in

decreasing order, as 7r < rj with i < j would imply that 7rj is not a left-to-right minimum.

Therefore, v' is composed of two decreasing sequences, so by the Pigeon-Hole Principle

there cannot be an increasing subsequence of three entries (three pigeons cannot fit into

two pigeon holes). So, we have mapped our 132-avoiding 7 to the 123-avoiding 7' and can

apply this process to all r S,,(132).

To see that this process is reversible, we fix the left-to-right minima of r', leaving

blanks at the other indices and placing the remaining entries in a sack. Moving left to

right, we fill each blank with the least entry still in our sack which is larger than the

rightmost left-to-right minimum to the left of said blank. Now we have a permutation kr,

which we claim to be 132-avoiding. Indeed, if there were a copy of 132, then there would

be a copy of 132 beginning with a left-to-right minimum, however this is a contradiction as

the entries following and larger than each left-to-right minimum are increasing. Thus, we

have a bijection. O

So, in fact we see that for all a, rT E S3 and n > 0, s,(a) = s,(7). One may be

tempted to suspect such a statement holds for patterns of every length m. However, with

a computer check or a few pages of scribbling, one obtains

S6(1342) = 512 / s6(1234) = 513.

We now turn to .,-i-.,il'uitics. In 1980, Richard Stanley and Herb Wilf independently

conjectured that for each pattern a there exists a constant c, such that, for all n > 0, we

have

sn(a7) < c.

In [2], Arratia proved that this was equivalent to the following, long known as the

Stanley-Wilf Conjecture.

Theorem 8 ( r! ircus-Tardos). Let a be a pattern. Then, the limit

lim 8,(a)1
n-*oo

exists (and is finite).

The validity of the Stanley-Wilf Conjecture was established by the 2003 proof by

Marcus and Tardos [23] of the Fiiredi-Hajnal Conjecture on permutation matrices. That

the Fiiredi-Hajnal Conjecture implies the Stanley-Wilf conjecture was proven by Klazar

[21]. For a clear and concise treatment of all, see section 4.5 of [5]. The reader is also

encouraged to see Doron Zeilberger's alternative rendition [41].

There is still hope for a tighter proof of the Stanley-Wilf Conjecture, as the

Fiiredi-Hajnal Conjecture only proves there exists a constant, but the constants which

we get from the proof are astronomically larger than the observed constants. It does still

give structure to our work to know that for any pattern a, s,(a) is at most exponential,

and additionally the limit

lim s,(O)l/
n-*oo

exists. We denote this limit by L(a) as it is critical to the sequel.

Now, for a finite set of at least two patterns, we do not have such a strong general

result. For any such set of patterns E, it is readily seen that s,(E) < s,(c) for each

pattern a E Z, so we have the following corollary to Theorem 8.

Corollary 1. Let E be a no", mnil;i finite set of patterns. Then, there exists a constant cy

such that for all n > 0,

Sn(Z) < CY.

So, for any such set of patterns E, we see that the sequence s,(E) is at most

subexponential, i.e. bounded above by an exponential function, but we did not mention

the existence of a limit of s,(Z)1/'. For many classes of such sets we cannot prove the

existence of such a limit, an open problem with much interest, discussed in [37].

Our work in this chapter is restricted to sets of the form {q, a}, where q is a

monotone increasing pattern, i.e. q = 12 ... r for some r > 3. We have the following

result on a class of such sets, a generalization of Arratia's proof [2] that the exponential

bound (i.e. the Marcus-Tardos theorem) implies the existence of the limit for the case of a

single pattern. This generalization was -ir-.- -1-- 1 by M. Klazar [22].

Definition 10. We -.';, a permutation 7r E ST is decomposable if there exists 1 < k < n

such that for all 1 < i < k < j, ri > Tj.

Lemma 3. Let q = 12 r and a E Sm for some r > 3 and m > 3. Then, if o is not

decomposable, the limit [

L(q, a) lim s,(q, a)1"
n-oo

exists.

Proof. Let m, n > 1. For each pair of permutations a E Sm(q, a) and 3 E S,(q, a),

construct the (m + n)-permutation

F (al + ((a + ) ( n) (.m.( + n) 1 0 3 2 /3.. .n

By our assumptions on a and 3, 7r avoids q and a in its first m entries and in its last n

entries. Furthermore, as q and a are not decomposable, there is no <" *i of q or a which

contains entries from these two sets. Thus, 7r is (q, o)-avoiding and uniquely determined by

a and 3, proving the following inequality.

Sm+n(q, o) > Sm(q, -)Sn(q, -).

We have the following lemma by Fekete [17] for superadditive sequences. The analog of

this lemma for subadditive sequences was used in Arratia's proof for the case of a single

pattern.

Lemma 4 (Fekete). For every sequence {a,}fn N '.:.''' ..'1

an+m > anam

for all m, n > 0, the limit

lim
n-oo 7

exists.

Applying Fekete's lemma to the sequence log s(q, oa), we thus have that the sequence

s,(q, -)1/" has a limit. As this sequence is bounded above, this limit is finite. D

With these facts in mind, we define L on sets of two patterns as follows.

Definition 11. Let a and r be patterns. Then, /. fI;,. the following ,I;il.:/l;

L(q, ) =lim sups,(,r)1 "'
n-*oo

Of course, for pairs of patterns a and r for which the limit exists, such as those in

Lemma 3, the limit and lim sup agree, and L is as we want it to be.

2.2 An Open Problem by M. Atkinson

This work was originated in response to a question posed by Michael Atkinson at

the fifth International Conference on Pattern Avoiding Permutations. We answer the

question in the affirmative for all increasing patterns q on some classes of patterns a and

specifically for q = 123 for larger classes of patterns a.

For a pattern a of length m, define a' to be the pattern (m + 1)a. For example, for

a = 2431, we have a' = 52431. Given a monotone ascending pattern q and a pattern a,

does it hold that L(q,a') = L(q, a)?

In any permutation the entries preceding the first ascendee form a decreasing

sequence. For a permutation 7r, if the first ascendee of 7 is the index i, we call Tr the

threshold of 7 and call the decreasing sequence preceding the threshold the front end. The

set of entries following the threshold we call the back end.

Example 9. For 7 = 532614, our first ascendee is the index 4, so the threshold of 7 is the

entry 7F4 = 6, the front end is 532, and the back end is 14, as shown in Figure 2-2.

Note that in our previous example each entry of the back end is less than the

threshold. Creating our own good luck, we chose 532614 for our example specifically

because it avoids 123. In fact, every 123-avoiding permutation shares this property,

a simple structure of which we shall take great advantage in our handling of these

permutations. To see this property, suppose our threshold is rt and there is an entry

rj in the back end (equivalently j > t) with Tr > ft. By definition of ascendee, it-1 < 7Tt,

and the subsequence 7rt_-1 t Tj forms a 123.

We borrow the next few definitions from the recent paper by Vatter [37].

* Threshold

Front End

Back End

Figure 2-2. The permutation 532614.

Definition 12. An interval in a permutation 7r is a set of consecutive indices

{i, i + 1,..., i +r} such that the set of values {1i, iTii, .. i+r} is a set of consecutive
integers. A decreasing interval is then a set of consecutive indices whose values appear

in decreasing order. F.:,,,ll, a maximal decreasing interval is a decreasing interval

{i, i + 1,..., i + r} such that {i 1,..., i + r} and {i,..., i + r + 1} are not decreasing
intervals. Increasing intervals and maximal increasing intervals are 1, 17,.,, ,..irl,l;, I.;,-1;

We denote an interval by its set of indices, written with set notation, or by its entries,

written in one-line notation.

Example 10. Let r = 3756124. Then, 7r contains the interval {2, 3, 4}, i.e. 756. Indeed,

{2,3, 4} is a set of consecutive indices, and the set of values {12, 3, } 4= {7, 5, 6} is a set

of consecutive integers. Note that 7r also contains the maximal increasing interval {5, 6},

i.e. 12.

Definition 13. Given a permutation w E S, with interval I = {i, i + 1,..., i + r}, we

/. I;,.' the deflation of 7 at I to be the reduction ofrl ri_1 i 7ri+r+l 7"r,. Similarly,

the inflation of 7 at the index i by the permutation a E Sm is the permutation obtained

from 7 by increasing by m 1 each entry greater than ri and replacing the entry Tr with

the interval whose reduction is a.

Example 11. The /. fl.,[/:.n of the permutation 264513 at the interval 645 is the reduction

is 2413. The :,fl/rl,.,n of the permutation 3124 at index 3 by the permutation 321 is

514326.

It was remarked above that the front end of a permutation is monotone decreasing.

Thus, the front end has a unique factorization as the concatenation of one or more

maximal decreasing intervals.

Example 12. Let 7 = -.. :21947. Then, the front end of T is 865321, the concatenation

of maximal decreasing intervals 8, 65, and 321.

2.3 Generating Trees

A tree is simply a connected, 1 i i': simple graph, or equivalently a connected simple

graph on n vertices with n 1 edges. A forest is a collection of trees. A rooted, labeled

tree is a tree with an assignment of labels (typically non-negative integers) to the nodes

and one node designated the root, giving an orientation to the entire graph. We -- that

a node y is a child of the node x if the final edge in the unique path from the root to y

is the edge {x, y}. Likewise, we call x the parent of y and define the depth of y to be the

number of children. The descendants of y is the set of nodes x such that the unique path

from the root to x passes through y. Finally, we call the set of all nodes whose unique

path to the root contains k edges the kth generation of the tree.

In [39] Julian West defines a generating tree as a rooted, labeled tree having the

property that the labels of the children of each node x can be determined from the label of

x itself. This leads to the characterization of a generating tree by the label of its root and

40

3

2

Figure 2-3. The permutation s.. :21947.

a set of succession rules which determine the number of children and labels of children for

each node of a given length and label.

The classic task for a combinatorial enumerologist is to determine the number of some

combinatorial objects of size n, perhaps further indexed with respect to some property or

some statistic k. Typically, one is presented with an initial object of some small size and a

recursion rule which z-,i- how many objects of each successive generation (objects of size

n + 1) can be created inductively from those of the previous generation (objects of size n).

Define the gth level-number of a tree to be the number of nodes in the gth generation.

Thus the generating tree is easily seen as a tool which lends itself quite readily to

combinatorial enumeration. We consider the nodes of our tree to be the combinatorial

objects themselves. There are many situations when the number of (n + 1)-objects which

root

0 *' children

Third generation of tree

Figure 2-4. A rooted tree.

can be generated from any n-object is all we need to know, so we might as well label each

node with its depth.

In [39] West begins with a trivial example, the complete binary tree. We begin with a

root with label (2). Our succession rule is that each node with label (2) has two children

also labeled (2).

Example 13 ([39], Example 1). The complete binary tree is determined by the set of rules

Root: (2)

Rule: (2) (2)(2)

A long celebrated integer sequence is that of the Fibonacci numbers (F,)n>_o

0, 1, 1,2, 3, 5,..., where we have the initial assumptions F0 = 0 and F1 = F2 = 1, and each

additional number F, is the sum of the preceding two numbers of the sequence. They will

1p1 i a role in our studies of pattern avoidance, so we show their generating tree as a less

trivial example of generating trees and a slow introduction to this sequence.

(2)

(2) (2)

(2 (2) (2 (2)

Figure 2-5. The complete binary tree.

Example 14 ([39], Example 3). The Fibonacci tree is determined by the set of rules

Root : (1)

Rules: (1) (2)

(2) (1)(2)
The observant reader undoub', l./; noticed that our succession rules are not the same as the
recursion we gave to 1. F,'.: our sequence. We verify that these rules are in fact equivalent
to the statement F~i+l F= + F_1-. For each n > 1 we have a set of G\$) objects labeled
(2) and a set of G\$) objects labeled (1). Each object in either set produces an offspring of
size n + 1 with label (2). So, we have

(2) G(1) + G(2) = .
Gn+l n n

Furthermore, each object in G,1) produces an offspring of size n + 1 with label (1), so

G(1) (2) n_ 1.

where the second ..,; ,'l/:/l follows from our previous statement. We have thus accounted for
all Fn+i objects, and we have our recurrence.

(1) (2)

(2) (1 (2)

Figure 2-6. The Fibonacci tree.

For a detailed exposition on the use of generating trees in the study of pattern

avoidance, see [39], [12], [36], [10] and [24].

Here we define the generating trees which will be used throughout. These definitions

depend on the patterns a and q which are being avoided, so we assume the patterns to

be given. This will be clear from context. First we explain the motivations. Recall our

/ notation. For a pattern a E S,, the pattern a' E S,+1 is obtained by prepending a

with the entry (m + 1). The fundamental question here is whether various limits (or

limit supreme) for the number of permutations which avoid some pattern a are the same

as those which avoid a' (assuming for now that the limits exist). It was noted above

that a-avoidance implies a'-avoidance, but there are a' avoiders which contain a. So,

our question boils down to just how many of these there are, in particular what are

the .i-,iii1ll 1.ics of these permutations with respect to the set of a-avoiders. We would

like to make use of generating trees to study the set of a'-avoiders which contain a, i.e.

{1 : w avoids a'} \ {w : avoids a}.

For each n > 0, let T, = T,(q, a) be the set of all (q, a)-avoiding permutations of

length n, enumerated by t,. We construct the generating tree T = T(q, a) whose nt level

is T,. A permutation 7 e T,,+ is a child of 7 e T, if and only if 7 can be obtained by

inserting n + 1 at one of the n + 1 open sites in 7. Similarly, for each n > 0, let U" be

the set of all (q, a')-avoiding permutations of length n, enumerated by u, and construct

the generating tree U = U(q, a) with levels U, and succession defined as for T. Atkinson's

question is thus answered in the affirmative for some pattern by showing that u, does

not grow ..-i ,i 11..I i. ly faster than t,. We will focus our attention on the sets W, of

all (q, u')-avoiding permutations of length n which contain at least one copy of a, i.e.

W, = Un\T,, enumerated by 1',,. This motivates the forest W = U\T. Note that each

tree in W is rooted at a q-avoiding permutation which avoids a' and contains a but whose

parent in U avoids a. For q = 123 and a = 132, we have the tree shown in Figure 2-7.

An active site in a permutation is a valid insertion point, that is, a site where we

can insert n + 1 and obtain a child which is still in the current generating tree, so for

our purposes an active site is such that the insertion will not cause an occurence of any

pattern which we seek to avoid. The depth of a permutation 7 is the number of active

sites in 7, equivalent to the notion of depth defined above on generating trees. We note

that the depth depends on both the permutation itself and on the tree, specifically the

pattern being avoided which determines the tree.

2.4 "Hat" Notation

Given a permutation 7 and a pattern a, we denote by a any copy of a in T. For

each index p, we use the notation ap to refer to an entry which serves the role of ap in

some a. As there may be more than one <". ,- of the pattern a in some permutation, ap

generally does not refer to any specific entry. For example, with a = 132, the permutation

r = 24135 contains one a, namely 243. In this case a1 refers to the entry 7t = 2, a2 refers

1 (2)

21 (3) 12 (2)

321 (4) 231 (1) 213 (1) 312 (2)

4321 (5) 3421 (1) 3241 (1) 3214 (1) 4231 (2) 4213 (2) 4312 (3) 3412 (1)

Figure 2-7. T(123,132).

to 72 = 4, and 63 refers to T4 = 3. On the other hand, the permutation 1432 contains
three -'s, namely 143, 142, and 132. In this case we can refer to the a1, the entry 1, but

we have several a2's and several a3's. It should also be noted that one entry could be both

a -i and a aj for some i / j.

Example 15. With a = 132, in the permutation 7 = 25431, r3 = 4 is the 2 = of the

subsequence 254 as well as the -3 2 of the subsequence 243.

Often we will determine the depth of a permutation 7 by the index of the rightmost

a1 in 7, by which we mean the entry with the greatest index of those entries rk such that

there exists a a which begins at rk.

Example 16. For a = 231 and 7 = 43521, the entries 71 and 72 are both I1 's, and we call

72 the rightmost i1.

Any confusion over hat notation should fade upon seeing its motivation in the

following proofs. As long as we use the ^ notation carefully, the meaning should alv--i- be

clear.

2.5 Monotone Increasing Patterns q

For any pattern a we have by construction that a' contains a, so it follows immediately

from our introductory remarks on pattern avoidance that

sn(() < s,,(t').

for all n, and

L() < L(-').

Of course, these bounds also extend to

L(a, n) < L(7', n).

for any set of patterns II, etc.

Let q be the increasing pattern 123 ... r. We will show that the statement L(q, a') =

L(q, a) holds if a pattern a begins with its greatest entry. Our proof builds on B6na's

proof for the case of single patterns, found in [4].

Definition 14. A left-to-right maximum is an entry in a permutation which is greater

than each entry to its left. The remaining entries of a permutation are those which are

not left-to-right maxima A weak class (weak n-class) is a set of permutations (resp.

n-permutations) whose left-to-right maxima are the same and are in the same respective

positions.

Example 17. The permutations 32415 and 31425 both have left-to-right maxima 3, 4, and

5, at the first, third, and fifth entries, so we C-,1 that 32415 and 31425 are in the same

weak class or weak 5-class.

Example 18. The permutations 3412 and 2413 both have left-to-right maxima in the first

two positions, but as the maxima themselves are not the same, 3412 and 2413 are not in

the same weak 4-class.

Lemma 5. For each r > 1, there exists a py. '1';. ;,,;. fr(x) such that for all n > 1 the

number of weak n-classes of permutations with ,.. /i;, r left-to-right maxima is less than

frn).

Proof. Fix n. We can easily count the weak n-classes with one or two left-to-right

maxima. An n-permutation with only one left-to-right maximum must begin with n,

so there is only one such weak n-class. Next we claim that there are (2) weak n-classes

with exactly two left-to-right maxima. Indeed, pick two numbers a, b E [n] with a < b.

Place the entry a in the first position, and place the entry n in the (n + 1 b)th position.

Given such constraints, we can ahv--li- place the remaining entries to find at least one

permutation in each weak n-class. For r > 2 we take the same approach but allow

overcounting. There are at most (/) 2 v--; to choose the entries and positions (numerical

and geographical values) for the r left-to-right maxima, as this includes all possible

sequences of left-to-right maxima, as well as some sequences which cannot possibly be the

set of left-to-right maxima of a permutation. As (7)2 is a polynomial in n for any fixed r,

the statement holds. O

Corollary 2. For each r > 1, the number of weak n-classes with less than r left-to-right

maxima is bounded by a ./, ;;.,;;,,.l in n.

Proof. For each 1 < k < r, we have a polynomial upper bound fk(n) on the number of

weak n-classes with exactly k left-to-right maxima. The sum over all such k is clearly an

upper bound for the number of weak n-classes with fewer than r left-to-right maxima. As

a sum of (a fixed number of) polynomials is a polynomial, we are finished. O

Proposition 19. Let q be the ascending pattern 1 2 ... r, and let a be a permutation which

begins with its greatest entry. Then,

L(q, a') L(q, a).

Proof. We first note that a permutation which avoids q has fewer than r left-to-right

maxima. Now, if a permutation avoids a', then its remaining entries avoid a. Indeed, by

definition each remaining entry is preceded by a left-to-right maximum. If the remaining

entries of a permutation contain a, then we may prepend this a with any left-to-right

maximum which is to the left of and greater than a1 to obtain a a', as ai is itself greater

than all other entries of a by the fact that a begins with its largest entry.

Therefore, we can overcount n-permutations which avoid a' by multiplying the

number of weak n-classes which have fewer than r left-to-right maxima by the number of

possible (q, a)-avoiding permutations of the remaining entries. By Corollary 2 the number

of such weak n-classes is at most a polynomial function f(n). Therefore our overcount of

n-permutations is f(n)s,_i(q, a). We are now in position to take our limits.

L(q,a') = lim sup s,(q, a')1//
n-*oo
< lim sup (f (n) s,-(q, ))1/T

-lim sup f(n)1/"s,_-I(q, a) 1/n

= lim sup 1 s_-l(q, a)1/n7

= L(q, a).

Combined with the knowledge that L(q, a') > L(q, o), we are finished. O

The following lemma from [4] and [5] provides an upper bound on the number of

permutations of length n which avoid the increasing pattern of length r.

Lemma 6. s,(1 2 .. r) < (r 1)2n for all r, n > 2.

Proof. We define a rank function on the entries of each permutation 7t E S,(12 .. r) by

setting the rank of an entry 7r to be the length of any maximal increasing subsequence

7i,17,2 ... 7i. Note that this generalizes the concept of the left-to-right-maxima, which are
precisely those entries with rank 1.

If an entry 7i has rank t, then there exists some increasing subsequence 7rjTrj2 -I rj1~Ti

of length t, so for any entry trk > 7Ti with k > i, the rank of trk is at leas t + 1, as we have

the increasing subsequence 7rj'Trj2, it-rj k. Therefore, for each 1 < t < n, we see that

the entries of rank t form a decreasing sequence.

There are at most n such decreasing sequences, and as sets of indices they form a

set partition of [n]. As 7r is assumed to be 12 .. r-avoiding, 7r has no entry of rank r or

greater, and there are at most r 1 blocks of our set partition. Assigning each entry of 7r

to one of r 1 blocks, we may overcount and see that there are at most (r 1)" possible

assignments of the indices to the blocks and at most (r 1)" possible assignments of the

values to the blocks. O

With this lemma in hand, we subtly alter another proof of B6na to achieve:

Proposition 20. Let q be the ascending pattern 1 2 .. r, let a be a pattern, and let c be a

constant such that for all n > 1, s,(q, a) < c". Then, for all n > 1,

Sn(q, U') < (c + (r- 2)2)-1.

Proof. In order for a permutation to avoid a', it is necessary that it avoid a in the region

to the right of n. So, we may overcount the number of (q, a')-avoiding permutations

by choosing where to place n, which entries to place to the left of n such that they are

1 2 ... (r 1)-avoiding (because any 1 2 ... (r 1) among them could be postpended

with n to create a q), and how to arrange the entries to the right of n such that they

are (q, a)-avoiding. We let k be the position of n, so there are ("- ) possibilities for the

entries preceding n. By Lemma 6 there are at most (r 1)2(k-1) possible permutations

of these entries. Finally, by our original hypothesis on S,(q, a), there are at most c"-k

possible permutations of the n k entries which follow n. Altogether, these work out to

the binomial expansion

s,(qo)') < (r 2)2(k-1)C-k'
k-l
k= 1

(c+ (r- 2)2n-1

concluding the proof. O

In particular, for q = 123, we have (r-2)2 = 1, so with the assumption s,(123, a) < c"

for all n, we find s,(123, a') < (c + 1)"-1 for all n.

The Stanley-Wilf Conjecture (il rcus-Tardos Theorem) tells us that for any pattern

a or set of patterns E there is such a constant c as in the above hypothesis. In the case of

avoiding a single pattern a, Arratia showed in [2] that the sequence s(o-)1/"' is increasing.

However, there are sets of patterns E for which the sequence s,(E)1/" is not increasing.

Thus, taking c to be the least constant such that s,(E) < c" for all n < N for some N, it

may be that there is a constant d < c such that s,(E) < d" for all n > N. In particular,

the constant c may be significantly greater than L(E), so the new constant d is closer to

our limit and thus a better indicator of the ..i-mptotic behavior of our sequence s,(E).

Such a situation motivates a strengthening of the previous proposition.

Proposition 21. Let q be the ascending pattern 1 2 ... r, let a be a pattern, and let d be a

constant such that for some N and all n > N, s,(q, a) < d". Then, there exists a constant

D such that, for all n > N,

s,(q, a') < D(d + (r 2)-1.

Proof. The set {s,(q, a) : 1 < n < N} is finite, so it is bounded above, and we can choose

a constant D such that s,(q, a) < Dd" for all 1 < n < N. We retain all machinery from

the proof of the previous proposition, except instead of counting at most cn-k possible

permutations of the n k entries which follow n, we count them by Ddn-k. Then our

expansion becomes

n t (r-
sn(q, a') < nj [ r- 2)2 k-1)Ddn-k
k 1

k 1
D t) (r 2)2(k-1)dn-k

D(d + (r 2)2)-.

2.6 The Pattern q 123

In this section we restrict our attention to the 123-avoiding environment. Some

statements will be generalized to longer ascending patterns q in the following section.

Consider the active sites of a permutation 7r W. As any child of 7 is 123-avoiding,

there can be no active site to the right of the first ascendee. As any child of 7 is

o'-avoiding, there can be no active site to the left of a &1. However, the consecutive

sites satisfying these two criteria are all active. So, our understanding of depth reduces

to understanding where these two bounds lie. Furthermore, if 7 has depth d, inserting

n + 1 into one of the d active sites will not increase the depth. Indeed, n + 1 is inserted

to the left of the first ascendee of 7 and itself becomes the first ascendee in the child.

The rightmost a1 of 7 remains in place in the child, so the child will have a rightmost a1

which is at the same position or to the right of that of T. This demonstrates the following

lemma. Recall our definition of the threshold of a permutation as the leftmost ascendee of

the permutation.

Lemma 7. Let 7r W for some pattern a. Then, the depth of 7 is the distance from the

,i.:1, i,,i ,-- 1 to the threshold, i.e. the difference of the index of the threshold and the index

of the rightmost a-1.

We proceed with a lemma which will be used extensively as it provides a polynomial

bound for the level-numbers of each tree in the forest W.

Lemma 8. Let a be a pattern of length m which does not begin with m. Let 7r be a

(123, a')-avoiding permutation with depth d which contains a but whose parent avoids a.

Then, the number of descendants of 7 at the jth generation, i.e. the jth level-number of the

tree in W rooted at 7, is bounded above by

(d +

which is a i ;, .. ;;,.:' in j of degree d.

Proof. (We are in fact bounded by the lesser polynomial (d+- 1), however we dropped the

-l's for neatness.) Let n be the length of Tr. First we note that 7r does not begin with n.

Indeed, as the parent of 7r has no a but 7r has a a, n is the rh in any a in Tr. 7r contains a

copy of a, which does not begin with its largest entry, so n appearing to the left of any a

would complete a a', contradicting the assumption that 7r is u'-avoiding. In fact, by this

argument we see that n is the threshold and a must contain entries in the front end of Tr.

We are trying to show that the number of descendants at each generation is bounded

by our polynomial, so we might as well assume the worst case scenario, that inserting at

the kth active site alv--i-i produces a child of depth k, i.e. that the rightmost -1 of a child

is alv--,v- at the same position as the rightmost a1 of the parent. Then, for all 1 < k < d,

we in fact have the succession rules

Root (d)

Rule (k) (1)(2)...(k)

An example of a tree in such a forest is shown in Figure 2-8.

42153 (3)

462153 (1) 426153 (2) 421653 (3)

4762153 4726153 4276153 4721653 4271653 4217653

Figure 2-8. Tree in W(123,231) rooted at 42153

Denoting by aj,k the number of permutations at the jth level with depth k, 0 < j and

1 < k < d, we have the recursive system

ao,d = 1

ao,k = 0 Vk / d
d
aj,k = Y aj-1,t Vj > 1, 1 < k < d.
t-k

From this recursive system we see that aj,k (dk+l+j) For each d and j we may

sum over all k to attain the level-number ( d+_l). However, a combinatorial proof is

preferable. We know that ( dj1) counts j-multisets of [d], and such a multiset written in

nonincreasing order spells out the order of active sites chosen in the lineage from 7r to a

permutation of length n + j. E

Definition 15. ([5] D. ii7.l' 5.33) A permutation 7 is called 1 i-, ,. if it can be written

as the concatenation qlq2 .. qk where each qi is a decreasing sequence of consecutive

integers and the leading entry of qi is smaller than the leading entry of qi+l for 1 < i <

k-1.

Example 22. An example of a 7.';,. ,, I permutation is 213654, which has three 7.';, ,-, as

shown in Fi:,,,n 2-9.
6

5

4

3

2

Figure 2-9. The 1 i, ,I permutation 213654 with lI-v. r- 21, 3, and 654

AT ii: results are known concerning pattern avoidance and pattern packing for 1 iv. I

permutations. One can see Section 5.2.2 of [5], [27], [6], and [20]. Our next result is on

1 ,i,, ,.1 patterns with just two I1V, rs, equivalently non-monotone 1 ,i ri patterns which

avoid 123.

Proposition 23. Let a be a pattern of the form

d (d- 1)...1 m (m 1)...(d + 1)

for some d > 1. Then,

L(123, a') L(123, a).

Proof. Note that any permutation which contains a has depth at most d. In fact, we will

use a stronger property of the roots of the trees in W in which the depth is exactly d.

Indeed, let the N-permutation p be such a root. We know that p contains at least one

copy of a, while the parent of p in U avoids a. As the only change between the parent

of p and p is the insertion of N, N must act as m in any a in p. This implies that there

is a set of d entries preceding N which serve as the d, (d 1),..., 1 in a a in p. In fact,

as the front ends of both a and p are decreasing (because a and p are 123-avoiding),

and as every front end entry of a is less than every back end entry of a, the d entries

.: I,, .:Vl,. /;/ preceding N serve these roles. Therefore, the rightmost o- cannot be more

than d positions to the left of N. As N is the only im in p, the entry which is d positions

to the left of N is clearly the rightmost o-1.

That there is a fixed depth which applies to every root in W makes our job easy.

For each N, the total number of trees in W whose roots have length N is less than

NuN-1, as each such root in W has a parent in U on the (N l)t level of U, and an

(N 1)-permutation can have depth at most N. Each such tree has level-numbers which

are bounded above by the polynomial (d+t-N-l1) So, the ^th level-number of WI, -i is

bounded by

I',, NUN-I
N= v
,a

N=m

(fd+n-
< d-

u, times a fixed polynomial f(n), a polynomial

that s,(123, a) < s, (123, ') for all n, we have

n- N 1
d-1

+n-N-1d

- 1 t-
n -1

which only depends on a. With the fact

The nth roots of this fixed polynomial approach 1, so we may apply the Squeeze Theorem

to achieve

lim sup '1/ = lim sup u1/'.
n-*oo n-*oo

We just saw that, if the front end is composed of entries which are all lower than

any entry in the back end, then there is a polynomial function which depends only on the

pattern and dictates the number of descendants for any permutation. Now consider how

the situation changes if there is more than one maximal decreasing interval in the front

end. Our shortest and prototypical example is 3142. The critical difference here is that

our depth is no longer bounded as it was for the patterns considered in Proposition 23.

For example, for any d > 2, we have the (123, 3142)-avoiding permutation (d + 1) (d -

1) (d 2) ... 1 (d + 2) d, which has depth d. This means we can are not guaranteed a

fixed bound to the degrees of our generating polynomials of the level-numbers of our trees.

All is not lost, however, as we can show that for any of a class of patterns a and any k

there is an upper bound to the number of trees in W whose root has depth which is k

less than its length. For the pattern 3142 it follows easily from the fact that the depth

of a permutation in W is completely determined by the length of the second maximal

decreasing interval in the front end. First, we calculate L(123, 3142).

As was mentioned in our introduction to generating trees, a famous combinatorial

sequence is the Fibonacci numbers F, defined for all n > 0 by the recursive system

Fo 0

F, 1

F,~+ = F, + F_ 1 V n > 1.

Multiplying both sides of the recursive equation F,+ = F, + F,_1 by x", summing

over all n > 1, and solving for the ordinary generating function F(x) = Zn>o Fnx", we

find

F(x) (() -

from which we see that F, is .i- mptotically approximated by the nth power of the

dominant term, the golden ratio,

F (1+

Furthermore, as the golden ratio enjoys the property of being a solution to the equation

2 = + so that ( )= 1 3+, we immediately achieve an approximation

for every other Fibonacci number

2F n

By iteratively applying the recursive equation for F,, we achieve a recursion on F2,-1:

F2n-1 F2n-2 + F2n-3

= F2n-3 + F2n-3 + F2n-4

= F2n-3 + F2n-3 + F2n-5 + F2n-6

SF2n-3 + F2k-1
k=0

S2F2n-3 + F2n-5 + F2n-7 + + F1.

Proposition 24. For all n > 1,

s,(123, 3142) = F2n-1.

where F, is the the nth Fibonacci number. Furthermore,

3+5
L(123, 3142) 3
2

Proof. As the number of (123, 3142)-avoiding permutations of lengths 1, 2, and 3 are

1,2, and 5, respectively, we see that the boundary conditions are satisfied. Writing sr for

s,(123, 3142), it suffices to show that our sequence satisfies the appropriate recursion,

n-i
Sn = Sn-, + Sn 2s,_1 + Sn-2 + + S1.
k=I

Given a (123, 3142)-avoiding permutation of length n 1, we can ahv--, prepend n, as n

can never be the first entry of a 123 pattern nor a 3142 pattern, so the new permutation

will remain (123, 3142)-avoiding. This is clearly an bijection between S,_1(123, 3142) and

permutations in S,(123, 3142) which begin with n, accounting for the first term, s_-l.

For a (123, 3142)-avoiding permutation 7r of length n which does not begin with n,

let j be the index of n, so 7j = n, (and j ranges from 2 to n as 7r ranges over all such

permutations). As 7r is 123-avoiding, n is in fact the threshold. Therefore, the front end

of 7 consists of one maximal decreasing interval. Indeed, if the front end contains more

than one maximal decreasing interval, then there exist entries in the front end ri and i1i+l,

with the entry (~i+1 1) in the back end, and the reduction of r ri+1 n ( 1) is 3142,

contradicting our assumption on r.

As the front end of 7 is an interval, the deflation of 7 at the front end is a (123, 3142)-avoiding

permutation r of length n j + 2. The largest entry of is of course k2 = (n j + 2).

Removing this entry, we obtain a (123, 3142)-avoiding permutation of length n j + 1

(which may or may not begin with its largest entry).

We claim that this is a bijection. Indeed, given a permutation 7 E S,_j+1(123, 3142),

the inflation of 7 at the first index by the permutation (j 1) (j 2) ... 1 is a

(123, 3142)-avoiding permutation 7 of length n 1. Inserting n at index j regains a

permutation in S,(123, 3142), and this is an injection. Such a construction over all j

accounts for the terms s,_l + + si in our recursion.

The limit L(123, 3142) = 3+ follows immediately from the above discussion of the

.,i-',i i. i cs of the Fibonacci numbers.

Proposition 25. Let a be the pattern 3142. Then, the limit

lim s,(123, a)1"
n-oo

exists and

L(123, a') L(123,a),

i.e. L(123, 53142) L(123, 3142) 2

Proof. Suppose p is the root of a tree in W and has depth d and length n. Recall that

being a root implies p contains one copy of a, but the parent of p in U is a-avoiding. We

will map p to a permutation of length n-d+1 by removing certain superfluous entries and

reducing. By superfluous we mean that there is a subset of the entries of p which do not

add any information to the identity of p in the sense that they are completely determined

by the other entries and the fact that p has length n and depth d.

Consider the structure of the front end of p. We claim that its factorization consists

of exactly two maximal decreasing intervals. Indeed, n is both a 4 and the threshold of

p, so there exist 3 and 1 in the front end and a 2 in the back end, so the front end itself

cannot be a decreasing interval and thus contains at least two maximal decreasing

intervals. On the other hand, suppose the front end of p contains three maximal

decreasing intervals II, I2, and 13. Then, we can choose indices i1 E II, i2 E 12, and

i3 E 13 and indices j < k such that pil > pj > Pi2 and pi, > k > Pi3, and these five entries

form a a', contradicting the assumption that p is a'-avoiding.

We will proceed by removing the superfluous entries, reducing accordingly at each

step. We know n contained no information, because its location is determined by the

depth d of p and the location of the rightmost a-. Remove n, ridding the permutation of

any a. We retain the first (leftmost) maximal decreasing interval, but we do not need all

the entries from the second maximal decreasing interval, as we can detect their presence

(or absence) from d. So, deflate this interval. We now have a permutation of size n d + 1

which we call the 1' *'I. /i /,: of p.

Definition 16. Given a root p of W, the permutation obtained by /. fl.[i':.,i the second

maximal decreasing interval of p and removing its l',,l, .- entry is called its prototype.

By this process, we map our roots of length n and depth d into the set of a-avoiding

permutations of length n d + 1 which contain at least two maximal decreasing intervals

in the front end and contain an ascent (we alv--x leave a 1 and a 2). To see that the

mapping is injective, let 7 be such an (n d + 1)-permutation. We will construct the

permutation of length n and depth d of which 7 is the prototype. Let p be the lowest

index in the second maximal decreasing interval in the front end of 7. Inflate 7 at p by

the decreasing permutation (d 1) (d 2) ... 1, obtaining a permutation of length n 1.

Finally, insert n before the (p + d l)st entry. As this reverse mapping holds for any

a-avoiding permutation which contains an ascent and at least two maximal decreasing

intervals in its front end, we can overcount them as follows. We know that Sk(123, 3142)

is F2k-1, F2k-1 < 3k for all k > 1, and F2 1 03, where 3 (3 + 5)/2. So, for any

N, the number of roots in W of length N and depth d is less than F2(N-d+1)-1 and, letting

k = N d + 1 be the length of the prototype of each root, they each have level-number

polynomial

d+j-l1 N k + + j 1
d-1 N-k+1-1
(N-k+)

(N- k+(n -N)
N-k

(n k).
N-k

We can now overcount all permutations in W by the roots of their trees and the

prototypes of the roots:

n-l n ( k\
<, < F2k-1 Y k
k=3 N=k+1
n-l
< F2k-12 n-k
k 3
n-1
< 3k2 -k
k= 3
n-1
<
k 3

S- 3).in

In the end, we see that indeed the nth root approaches 3 = L(123, 3142). D

As was mentioned prior to this proposition, the pattern 3142 makes our work easier

because the front end of the pattern itself is the concatenation two maximal decreasing

intervals. The above proof provided a warmup for the main result of this section. The

following lemma allows us to work with longer front ends.

Lemma 9. Let a E Sm be a pattern with a decreasing back end. Let p E S, be a root of

W. Then, the front end of p has at most m maximal decreasing intervals.

Proof. We first note that if {i, i + 1,..., i + r} is a maximal decreasing interval in the front

end of p, with i > 2, then by maximality and the fact that the front end is decreasing,

pi + 1 is in the back end of p.

Assume the front end of p has at least m + 1 maximal decreasing intervals. We show

that this implies the parent of p in U contains a, a contradiction.

Label the rightmost m + 1 maximal decreasing intervals in the front end of p by

II, 2,... Im in order of their greatest entries. So, the front end of p is the concatenation

pl I.m+1mlm-1 _l- I1 or m1+11, II if p C ,m+1.

For each 1 < i < m, if the entry i is in the front end of a, we choose an entry in Ii

to be our i. If the entry i is in the back end of a, we choose an entry in the back end of p

which is greater than every entry of Ii and less than every entry of Iji+ (if i < m) to be

our i. Doing so, we achieve our 6. O

Example 26. If a = 231, and p = 97642531, then our four labeled maximal decreasing

intervals are II = 2, 12 = 4, = 76, and 14 = 9. Thus the front end of p is 14131211.

We construct our 6. As the entry 3 is greater than every entry of 11 and less than every

entry of 12, we choose 3 as our 1. Next we choose 4 as our 2. F.:,iull as 5 is greater than

every entry of 13 and less than every of 14, we choose 5 as our 3, and 453 indeed reduces

to a= 231.

We now answer M. Atkinson's question in the affirmative for patterns a which avoid

123 and contain the entry 1 in the front end, noting that this restriction implies the back

end is decreasing. Any such pattern is not decomposable, so we do know that the limit L

exists and the sequence s,(123, o)1/"' is non-decreasing.

Theorem 27. Let a E Sm be a pattern with the entry 1 in its front end. Then,

L(123, a') = L(123, a).

Proof. We generalize the proof of Proposition 25. Let do denote the depth of a. Note that

any root in W contains in its front end a (< li' of the front end of a and therefore has

depth which is at least do. Let d > do. We begin with a pl, l. ;, l'.' 7, a permutation in

Sk(123, a) for some k, and insert d do entries in 7 to construct a root in W with depth d.

Let 7 e Sk(123, a) have a child p(0) in U which is a root in W and has the same depth

as a, do. So, there exists an index t such that inserting k + 1 at t in 7 gives us p(o), and

7 contains a copy of \ m. If d = do, we take p(0) to be our root in W. Otherwise, we

build p(d-do) inductively. Let r be the index of the rightmost -1 of p(o), and note that by

construction t is the index of the threshold of p(o), and 7172 7 t-1 pl P2 pt-1

Besides the threshold, we will insert each entry using one of two methods. One

method is to insert at a maximal decreasing interval, i.e. choose an index in a maximal

decreasing interval in the front end of p() and inflate the permutation at that index by the

permutation 21.

Example 28. Let p() = 6431752. We i,,~r insert an entry by :ifl,i,; p(0) at index 2 to

obtain 75431862, where the '1,..1-fl.,.. 5 is the inserted entry.

The second method may be applied for any pair of entries in the back end of a which
(0)
have consecutive indices and consecutive values. Suppose pp) and pn+ correspond to such

a pair in the (".li, of a in p(0) and are the least such pair. By being the least such pair we

mean that if p = q, and the greatest entry in the front end of a which is less than aq

is ai, then the maximal decreasing interval containing ai contains the greatest entry in the

front end of p(0) which is less than dq.

As the depth of p(O) is the depth of u, there is no entry in the front end of p(O) whose

value is between those of p() and p1, so we have p) be the greatest

entry in the front end of p(o) which is less than p +1. Then, our second method of insertion

will be to insert the entry p(0) at the index i and increase by one every entry which is at

least as large as p().

Example 29. Let a = 41532 and p() = 521643. We i,,mii insert an entry in the front end

of po) to obtain 6421753, where the bold-faced 4 is the inserted entry.

Of course, in p(O) the pair pp and (0) have consecutive values, and this may not

be the case in p(j) for greater values of j, as we may have already inserted at an index

q using the second method and corresponding to the pair pp(o) and p 1. In this case, we

inflate p() at the index q by the permutation 21.

By Lemma 9, we know that the front end of p(O) has at most m maximal decreasing

intervals. In fact, there are exactly M maximal decreasing intervals in the subsequence

yr+1 .. 7t-1 for some M < m. Suppose there are P pairs of entries for which we may

apply our second method above, so altogether we have M + P choices for each insertion,
and we order them as C1, C2,..., CM+P. So, let {xi,... Xd-do} be a multiset of [M + P]

with xi > x 2> > Xd-do. For each 1 < j < d do, inductively define p(J) to be

the insertion into p(-1) determined by the choice Cx,. So, we obtain p(d-do) by inserting

entries into some of the maximal decreasing intervals of p(o). As these entries were inserted

between the rightmost o1 and the threshold, we have that p(J) has depth do + j for each j,

and each one is a root in W. Altogether, we obtain our root of depth d, p = pd-d)

Now, we have our root and we can overcount the (123, c')-avoiding permutations of

length n. We are in the same situation as in the proof of Proposition 25 except that we've
made (M+P+d-do-1) choices of how to insert the entries into our prototype to obtain our

root. Of course, M + P < m and d do < n, so this is bounded above by (m-n ). Note that

since a contains the pattern 132, L(123, u) > L(123, 132) = 2. Altogether, we have

<' <

m
< (mn +
m

< (m +
m

)n-l n
Sk(123, a) N
k 3 N= k+
nl
r) sk(123, a)2-k
k -3
n-1

k -3

n (n 3)s (123, e).

Taking the limit,

lim W <
n-noo
n-*oo

nlim m +n ( 3)1l/s,(123, a)1/
n-oo rm

lim sT(123, ))1"
n- oo

L(123, 7).

Experimental results support the following conjecture of Atkinson.

Conjecture 30. Let a be in,; pattern. Then,

L(123, 7') = L(123, 7).

CHAPTER 3
PATTERN PACKING

3.1 General Pattern Packing

In the previous section we posed anew some classic questions in the field of pattern

avoidance with the restriction that all permutations considered are given to avoid some

increasing pattern q. Along the same line, we can consider the question of pattern packing

on the set of permutations which avoid q.

We define the function pat on all permutations by letting pat(r) denote the number

of distinct patterns contained in the permutation 7. (Note that we i- distinct to avoid

confusion with the number of unique patterns, an entirely distinct (and unique) area of

research.)

Example 31. We have pat(1432) = 7, as 1432 contains the patterns 0, 1, 12, 21,132, 321, 1432

and no others.

An easy result on the function pat follows.

Proposition 32. Let q be the monotone increasing permutation 12... r. Then,

pat(q)= r +1.

Proof. As q is i i i -ii:.- so is every subsequence, so the only patterns we find in q are the

increasing patterns of length 0, 1,..., r. E

In the previous chapter we discussed equivalence classes for pattern avoidance. If the

permutation 7 contains the pattern a, then 7-1 contains a-1. Similarly for rR and 7'.

From this we immediately achieve for all permutations 7

pat(Tr) = pat(uT-) = pat( R') =pat( T).

So, 7, 7r-, 7 and 7r are in the same equivalence class with respect to the function pat.

Next, we define the function maxpat over all nonnegative integers by

maxpat(n) = max {pat(}) 1 E S, }.

An obvious upper bound to maxpat(n) is the total number of subsequences of a

permutation, 2". We are therefore interested in the i~mptotics of maxpat(n), or the

growth rate of maxpat(n)1/" as n -i oo. This question was originally posed by Herb Wilf

at the Conference on Permutation Patterns, Dunedin, Otago, New Zealand in 2003. He

presented the following class of permutations which give a lower bound for maxpat(n).

To construct Wilf's class, begin with the empty permutation, which we label W(o),

and the permutation 1, which we label W(1). Inductively assume that we have constructed

W("-1). If n is even, we postpend W("-1) with n, i.e. W(t) = W(-l)n. If n is odd, we

increase each entry by one and postpend W("-1) with 1. The first few permutations of our

class are evidently

W(1)= 1,

W(2) 12,

W(3) 231,

W(4) 2314,

W(5) = 34251,

W(6) 342516.

See Figure 3-1 for an example. The claim is that for all n > 1, pat(W(")) > F,,

where F, is the 'th Fibonacci number, as defined in the previous section.

Proposition 33. (Wilf) For all n > 0, we have the lower bound

maxpat(n) > F,.

Furthermore, we have the bounds

1+ /-
S < liminfmaxpat(n)1 / < limsupmaxpat(n)1 < 2.
2 n--oo --0o

3W

2

Figure 3-1. The permutation W(6) = 342516

Proof. We have pat(1) = 1 = Fi and pat(2) > 1 = F2, so the first statement follows

immediately for n = 1 and n = 2.

Now, for each n > 3, the number of distinct patterns we obtain from subsequences

of length at least 2 which end with W,) is equal to the number of distinct patterns in

W("-1), as any such pair of subsequences are of the form aWli ) and 31Wp ,) where a

and 3 are necessarily distinct subsequences of W("-1). Therefore, the number of distinct

patterns which we find in subsequences ending with the last entry of W(") is pat(W("-l)),

which is at least F,_1 by induction.

Likewise, the number of distinct patterns of length at least 2 in W(") which end with

W,1\$ is at least pat(W(-2)) > Fn-2 by similar arguments. It remains to show that these

two sets are disjoint. We note that, for n even (odd), each subsequence which ends at

W,) ends at its greatest (resp. least) entry, whereas each subsequence which ends at W )

ends at its least (resp. greatest) entry. These are mutually exclusive conditions by our

assumption that each pattern has length at least 2.

We see that indeed we have a lower bound which satisfies the Fibonacci recurrence.

From our previous work, we know that

F, -= v .5-

The upper bound on the limit supremum is trivial for maxpat, which is necessarily

bounded above by the number of subsequences of an n-permutation, 2". O

Based on empirical evidence, (see sequence A088532 in the On-Line Encyclopedia of

Integer Sequences [32]), it seemed this number may 1Ilio .... !i" the trivial upper bound of

2" (the number of all subsets of an n-permutation). In [15] this author constructed a class

of permutations over which it was shown that

lim maxpat(n)1" = 2.
n-*oo

However, while confirming our suspicion that the nth root approaches 2, this result

still left open the possibility that m axpn) 0 as n -- oo, i.e. the possibility that

maxpat(n) = o(2"). Recently Miller [25] and Albert et al. [14] independently proved with

a refinement of the original class from [15] and more delicate counting techniques that

indeed

lim =axpat() 1, i.e. maxpat(n) ~ 2 .
n-oo 2n

In particular, in [25], Miller showed the wonderfully exact bounds

2" (r22- ) < maxpat(n) < 2- (n2"- .

Definition 17. We generalize the function maxpat by 1. /I,.':.'/ for ,:1,; pattern a, the

function

1,,,., ,,,I(n) = max {pat(r) : E ST,(a)}.

Our task is to construct a class of permutations which maximize as much as possible

the function ,,,., .,/, (n) for increasing patterns q. First, consider what sort of upper

bounds we can find on i,,,,1 i i I (n) for a few simple patterns a.

Proposition 34. For q = 123, we have the upper bound

11'.uI(n) < t(n) in (Cm, ( ,))
0n o

where cm is the mth Catalan number and counts 123-avoiding permutations of length m.

Furthermore, we have the limit

lim t(n)/" = 2.
n-*oo

Proof. For each 0 < m < n, the number of m-patterns in any n-permutation 7 is at

most the number of subsequences of 7 of length m, i.e. the number of m-subsets of [n],

('). Similarly, there are at most Cm possible 123-avoiding m-patterns. As 7 itself is

123-avoiding, we know that any pattern contained in 7 is also 123-avoiding. So, for each

m, the number of m-patterns satisfies both these bounds, and the total number of patterns

is at most the sum of the minima over all m.

The significance of the above limit is that one may hope to pack just as many

patterns into 123-avoiding permutations as in the general case. Furthermore, for any

increasing pattern q = 12 ... r with r > 3, the same upper bound holds, albeit trivial.

3.2 Pattern Packing in 123-avoiding Permutations

Proposition 34 gives us hope that, even with the 123-avoiding restriction, we

may pack as many patterns as we would like in a permutation of sufficient length.

Experimentation supports such a conjecture. Here we give constructions for q-avoiding

permutations with in ,iy" patterns for increasing patterns q. Our constructions are

modeled on those of [15],[16], [14], and [25] and meet or surpass the original lower bound

given for the general case by Wilf, (1 ). We note that Wilf's original construction is

132-avoiding, so we already have it established that maxpatl32(n) > ( I )n.

We begin with a family of 123-avoiding permutations which show that the limit

infimum of maxpat123(n)1/" is at least 5 a n -- Oc.

We define our permutations P(") inductively. Let P() = 1 and p(2) 12. For each

odd n > 3, let P(t) nP("- ), i.e. p("-1) prepended with n. For each even n > 4, let P(n)

be p("-1) with n inserted immediately after 1. So, we have

P(1) 1,

p(2) 12,

p(3) 312,

p(4) 3142,

p(5) 53142,

p(6) 531642.

So, each P(") consists of two decreasing subsequences, and thus avoids 123. The next

proposition gives a lower bound on pat(n) with a proof similar to the proof given for

Proposition 33.

Proposition 35. For all n > 1,

pat(P(')) > F,.

Furthermore, we have the bounds

1 + < liminfmaxpat123(n)1/' < limsupmaxpati23(n)1/' < 2.
2 n__oo 0__o

Proof. Our induction hypothesis will be the stronger statement that P(") contains at least

F,1- patterns cor;lr.',,.:'. the entry 1. We have that P(1) contains the pattern 1, and p(2)

contains the pattern 12, so the statement of the proposition holds for n = 1 and n = 2.

Assume the statement holds up to and including n > 2. Then, P('+l) contains a set A

of at least pat(P(")) > F,_- patterns which contain the entry (n + 1) and the entry 1.
p(n+1) also contains a set B of at least pat(P("-1) > F_-2 patterns which do not contain

the entry (n + 1) but do contain the entries n and 1. We claim that the sets A and B are

disjoint. Indeed, if n + 1 is even (odd), then for each pattern in A the largest entry occurs

after (resp. before) the entry 1, while for each pattern in B the largest entry occurs before

(resp. after) the entry 1. Therefore, we have p(++1) > F,_1 + Fn-2 = F.. The bounds

3.3 Pattern Packing in q-avoiding Permutations

Next we extend our construction to longer increasing patterns q. Suppose q =

12 ... (r + 1) is the increasing pattern of length r + 1. We define the permutations Q(")

inductively. Set Q () 1, Q(2) = 12, ..., Q(r) = 12.. r. For n > r, our permutation

Q(") consists of r maximal decreasing subsequences. If n + 1 = p (mod r) with 1 < p < r,

we construct Q((+I) by inserting n + 1 at the beginning of the pth maximal decreasing

subsequence.

Example 36. If q = 12345, i.e. r = 4, we have

Q(5) = 51234,

Q(6) = 516234,

Q(7) = 5162734,

Q() = 51627384,

Q(9) = 951627384.

Definition 18. For k > 2, the Fibonacci k-step numbers, denoted FT ) for n > 0, are

, 1 [ .1 by

Fk)= 0 V n < 0,

F(k) .. = (k) =
I k 1

and, for all n > k + 1,

F (k) (k)+ F(k) + ... F(k)
,n n-k n-k+1 n-1

The classical Fibonacci numbers are the special case k = 2.

It is well known that the limit

lim (Fk)) 1
n- OC00

is the unique real root greater than 1 of the equation

Xk x x-2 + x 1.

This root, which we denote by ak, is called the k-anacci constant. We have that ak

increases with k and

lim k = 2.
k--oo

Proposition 37. Let q be the increasing pattern of length r + 1. Then,

pat(Q(")) > F r)
n-r+2"

Furthermore, we have the bounds

or < liminfn,,,l .,i, (n)1/ < limsup i,., i.,l (n)1/" < 2.
nfoo l*l00

Proof. For each 1 < n < r, we only count the pattern Q(") =1 ... n itself and have

pat(Q()) > 1 > F(r)
n-r+2"

For n > r + 1, we count the patterns of subsequences which contain the entries 1, 2,..., r,

i.e. subsequences which contain the lowest entry of each maximal decreasing subsequence

in Q("). Fix n > r + 1 and assume by induction that, for all N < n, the number of such

patterns is at least F(r) r2 For 0 < i < r 1, let Ai be the set of subsequences of Q(n)

which contain the entries 1, 2,..., r and whose greatest entry is n i, and let Bi be the set

of patterns of subsequences in Ai. We claim that the Bi's are pairwise dl-i- ~iil Indeed, for

i / j, suppose ai E Bi and aj E Bj, where n i p (mod r) and n j q (mod r),

0 < p, q < r 1. Then, the greatest entry of ai occurs after p ascents, and the greatest

entry of aj occurs after q ascents.

Also, for each i there is a bijection between the patterns in Bi and the patterns

counted for Q("-')-l. So, if we count all patterns in the union of the Bi's, we have

pat(Q )) > pat(Q "-)) + +pat(Q(n-r)).

As we have the recursion for the Fibonacci r-step numbers, the induction follows. O

It may be noted that these classes of permutations resemble the classes used in [15],

[16], [14], and [25], except that whereas in each of these papers the construction consisted

of k rows of k entries each, or a stripped down version of that, the permutations we are

using here have restricted row lengths (number of maximal decreasing subsequences) due

to the q-avoiding restriction.

It was noted that ak -- oo. From this fact we see that as we let r oo, our

constructions for increasing patterns of length r + 1 ipl ... !i" the constructions for the

general case and the lim infs approach 2.

3.4 Packing Density and Further Directions

Pattern packing is in a weak sense a dual concept to that of pattern avoidance. We

i- the duality is only weak as pattern avoidance, when compared to the total number

of permutations, can be understood as the p' '.1,1, l.:1.:1, that a permutation will have the

property that it avoids the given pattern, whereas the question of pattern packing is

extremal, asking what is the maximum of a certain statistic, the number of patterns

contained in a permutation, over all permutations.

Perhaps the proper dual concept to pattern avoidance question is a question also

posed by Herb Wilf, at the 1992 SIAM Conference on Discrete Mathematics. For a given

pattern a, how many copies of a can a permutation contain? The answer depends on the

packing 1. ,, .:/1 of the pattern a.

Definition 19. The packing density p(a) of a pattern a E S, is 1. I;,' ., by

g ((a, n)
p(a) lim -g n
n-oo (7)

where g(a, n) is the maximum number of copies of a in ,:n;, n-permutation.

For more on the work in this area, please see the Ph.D. dissertations of Dan Warren

[38] and Alkes Price [27]. Herb Wilf has recently -,-. -1i ,1 that the notion of packing

density be examined in the q-avoiding environment.

Why do we raise the issue of these dualities? We saw in the section of pattern

avoidance that in the restricted environments of 123-avoiding permutations, or more

generally q-avoiding permutations, it is more likely that a randomly chosen permutation

avoids some pattern T, or, critically, the chance of avoiding a approaches or equals the

chance of avoiding a', a statement which is usually false in the general case. So, restricting

to q-avoiding permutations makes our life easier in that work.

However, pattern packing becomes more difficult. Indeed, if we consider all approaches

used to prove pattern packing or superpatterns, they take advantage of a lattice or

checkerboard structure in the class of constructions to bound maxpat. If we restrict the

length of the increasing sequences allowed, we lose this structure. So, the meta-question

is whether we lose our .,-1ill ic limits or simply need more delicate techniques to see

them.

CHAPTER 4
ASYMPTOTIC NORMALITY AND UNIFORMITY

4.1 Probability Theory

In this paper we only deal with discrete probabilities and their limits. For the sake of

simplicity we avoid the measure theoretic foundations of probability theory and define a

smaller class of random variables, giving associated properties which we will need in this

chapter.

Definition 20. A random variable X is a function from the unit interval [0, 1] to the

real line R. For each r E R, we denote by P(X = r) the Lebesgue measure of the set

{w E [0,1] : X(u) = r} and call this the probability that X = r. The set of values

{X(w) : E [0, 1]} we call the range of X. A random variable X is called discrete if its

ri,.,,. is countable or if X has the weaker condition that there is a countable subset B of

the reals with P(X e B) = 1. F.:,,,ill; a 0-1 random variable or indicator random variable

is a random variable whose ru,,I., is the set {0, 1}.

Definition 21. Let X be a random variable with finite r,.ig R. The mean or expected

value of X is I,. ./ by

E(X) = r.
r 'ER
The variance of X is E((X E(X))2), which, by the i. ,,,, i of expectation, is also given

by

Var X = E(X2) (E(X))2.

The square root of the variance is the standard deviation.

We note that for a 0-1 random variable I, 12 = I, a fact that eases computation

of mean and variance for such variables. Indeed, given a 0-1 random variable I with

P(I = 1) = p, we have E(I) p and Var I = p p2.

As we will only be dealing (at least before taking limits) with random variables with

finite range, we will not develop all the measure theory behind these definitions. For this,

the reader is referred to the texts by Halmos [18], Taylor [35], and Chiing [13]. For a

treatment of discrete probabilities, the reader is referred to the texts by Alon and Spencer

[1] and CI', 1 .,, i1,. [11].

4.2 Triangular Arrays

Definition 22. Given a random variable Y whose rr,,,'. R is a finite set of integers,

1, i,'.: its probability generating polynomial to be

py(x) = P(Y r)xr.
rER

By a 1, .:.,,i,,,l,;r i ,,,r, of nonnegative real numbers (a,,k) we mean a sequence of

numbers which are indexed by n = 0,1, 2,..., and k = 0,1,... m m(n) for some

function m defined on N, so for each fixed n there is a finite number of terms an,k. The

term triangular array comes to us from the probabilists. When the sequence consists

of non-negative integers, we use the term combinatorial distribution found in [19]. In

combinatorial applications an,k counts objects of size n with some statistic k. For example,

in the next chapter we will study the numbers S,,k, counting set partitions of an n element

set into k blocks.

Given such a sequence an,k, we set Sn = an,i +an,2 + .. +an,m for each n and construct

a new sequence bn,k = a interpreting bn,k as the 1 /d, I.:./.; that a randomly selected

n-object has statistic k.

Example 38. A fundamental combinatorial distribution is that of the binomial coefficients

(~). It is well known that for each n we have the sum ( ) + () + + () = 2'. The

generating i .l;,,'... ;,.:'l
(x + 1) T () k
k-0
can be interpreted as the 1"' al.,,d.:.I/;, generating / .l;,,,;...;,.: for a random variable X, which

is the sum of n coin flips, i.e. X, is the sum of n independent 0-1 random variables Y,,k,

k 1, 2,... ,n, with
1
P(Y.,k 1)
2

for all n and k.

The term triangular array can be seen in the well-known Pascal's triangle listing the

binomial coefficients:
1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

We will only deal here with triangular arrays whose generating polynomials have

real, non-positive roots only. The motivation for this will become clear when it is seen

what strong statements can be made concerning such sequences and the clever, powerful

techniques used to deal with them.

4.3 Asymptotic Normality

For a sequence of random variables X, such that, for each n, X, is the sum of

m = m(n) independent, identical random variables, with m -- oo, the associated

distribution approaches the normal distribution, most easily understood as the limit (in

distribution) of the binomial coefficients. M ii;: significant results in probability theory are

of the form of Central Limit Theorems proving sufficient conditions for convergence to the

normal distribution.

Definition 23. For a random variable X, let X denote the normalized random variable
X-E(X) For a sequence of random variables X,, we write X, -- N(O, 1) to mean that X,
\Var(X)
converges in distribution to the standard normal variable. F.:,ill; we -.'; that a sequence

of random variables X,, is .,-i-','1iii 1 ically normal if X, -- N(0, 1).

We have the following theorem, appearing as Theorem 6.7.11 of [35].

Theorem 39 (Central Limit Theorem for Triangular Arrays). Given a t, :.,i!,sl., r

r ,,1; of random variables X,i~,..., Xn,m with respective variances o-n,,..., X,,m, let

a,2 o + + o-T, and Xn X,, + + X,,m. Then, the distribution converges

;i, l./; to the unit normal distribution if the Lindeberg condition is .,i/.:-f7 ,

Ve > 0 z x2Qk (dx) 0 as n oo,
k=1 Xn >,r|7n

where, for each k, Qn,k is the cumulative distribution function of Xn,k.

4.4 Asymptotic Uniformity

Another ..i-i !l- l ic property of some combinatorial distributions which provides us

with both qualitative and quantitative understanding of their ..i-mptotic nature is the

concept of .- -mptotic uniformity. We build up to this with B6na's definition of balance

([7], [9]).
Definition 24. For each positive integer q, we -.r;, that the combinatorial distribution an,k

is q-balanced if for each 0 < r < q, we have the limit

lim Zkr an,k 1
n-oo k an,k q

where all congruences are modulo q.

So, if aT,k counts n-objects with statistic k, q-balance implies that there is ;. ,.,lli

equal probability of a randomly selected n-object to have k-statistic in any one of the

residue classes of q.

Example 40. We give a quick proof that the binomial coefficients are 2-balanced. Evalu-

ating their generating r. I;,,/'; ... ,. (x + 1)" at x = -1 gives us the difference of binomial

coefficients for even and odd values of k:

(-1+1)" + (-C)kkC
k even k odd

As (-1 + 1)" = 0" = 0, we have that for all n > 1 the set of evens and the set of odds each

account for ir. I.I. half of the total.

Of course, most cases are not so clean as this and we must examine the limits.

However, we will see that evaluating the generating polynomial at the qth root of unity as

above is integral to our work.

Definition 25. We -.,r; the combinatorial distribution a,,k is i-mptotically balanced if it

is q-balanced for all q E P.

The following lemma and proof are based on those of B6na in [7] and [9]

Lemma 10. Let q E P. Suppose the combinatorial distribution an,k has generating

j,, ,J,, ,' !,,.,1d pn(x) /, ,'. ,/ by

Pn(x) -Z an,,x.
k-0
Then the sequence na,k is q-balanced if for all 0 < t < q

lim P. = 0, (4-1)
n-oo p,(1)

where ( = exp a primitive qth root of ii.l

Proof. Let 0 < r < q 1. For now we assume q divides m. Consider the sum

q-1
s,,,(() := Yp((t)-"r (4-2)
t=0

S( ak) -t (4-3)
t=0 k=1
m q-1
Ya. ^ Y (4-4)
k=1 t=0
m q-1
Z ,, (Y-) (4-5)
= 1an,k -5

Enclosed in parentheses in (4-5) is a geometric sum, and we take advantage of the

property that the (k r)-roots of unity add up to 1 to see that

1 O if (k-, .

t=0if k-r

Therefore, we have annihilated all terms in (4-5) for which q does not divide k r, and

(4-2) reduces to

m/q
S,r(() q an,jqr, or (4-6)
j=1

s,,(C) Zj1 an,jq+r
qp ) P (47)

We recognize the right hand side of (4-7) as the function from the definition of q-balance.

So, q-balance is equivalent to this expression converging to for all r, and assuming (4-1)

we can drop our assumption that q divides m, and we have the limit

sin zr (- 1-~)lit -1 ) -tr
nz-o qp. (t) q n^00 ^ P(1

q = n-0oo p(l)
q-1
1 -tr lir aP )
q toP

= (1 + 0 + + 0)
q
1
q

4.5 Generating Polynomials with Real, Non-Positive Roots

Throughout this section we will assume that a,n,k is a triangular array of non-negative

reals whose generating polynomials have real, non-positive roots and set s, = a,,o + a,1i +

.. + a,m,. Also, X,z will refer to the random variable defined by

P(X, = k)= a1k V k > 0,
Sn

with probability generating polynomial

Pn(X) = ank
k=0 Sn

The notion of .- I i llic normality or normal convergence is that the normalized

sequence _- converges in distribution to the normal distribution. For each n we will

find a set of 0-1 random variables whose sum is X, and use these to prove ..-i-!,i,;,il''

uniformity. We develop here the method attributed to Harper [19] and also used in [30].

By our assumption on the roots of our generating polynomials, we can factor the

probability generating polynomial pn(x) as

Pn (X) a H (x + Ank) an,,,m x + A,k
Sl
with 0 < A1 < A2 < .. < Am c R. For each n, define the mutually independent 0-1

random variables I,ji,..., In,m, by

P(In,k 0) = A,k and
1 + An,fk
1
P(In,k 1) t +,,k
1 + An,k

From this the mean and variance of each ,n,k follow immediately.

1
E(In,k)
1 + An,k

Var Ink E(Ik) (E(In,k))2
1 1
1 + An,k (1 + An,k)2
Arn,k
(1 + An,k)2

We have the respective probability generating polynomials gn,i(x),..., gn,m(x) given by

gn,k(x) P(I,k = 0) + P(In,k = )x n ,k+ X.
1 + An,k 1+ An,k

Now we can rewrite each probability generating polynomial pn(x) in terms of the

probability generating polynomials gn,k(x).

,() = an,m H x + A n,k
1
an,mn H gn,k(x)
l
We have thus expressed X, as the sum of the 0-1 random variables In,1... ,-, n,m. We

will be interested in the variance of X,, which we denote o2. We are now in position to

state the following theorem of Bender [3] which follows from Theorem 39:

Theorem 41 (Bender, 73). Let the combinatorial distribution aT,k be as above. Then, the

sequence of random variables X, is r- ;,/*i1.I..'.ll//; normal if and only if

an 00c as n oo.

We have the following computation, where 1 < j, k < m for all j, k.

2 = E(XX) (E(X))2

E((Z I. k) E(Z (I,k))2
k k

SE(Ink) + 2 E(IjIk) 2 2E(I -)) EI I
k j
Z= E(I,k) Z(E(,))2
k k

S Var ,k
k
n,k
(1 + Af,k)2*

With Theorem 41 and keeping the above notation, we have shown the following.

Lemma 11. With real, non-positive zeros, r/,,,, l...:'.' non,,'l.Ui is equivalent to the

divergence
im = c0.
noo (1 + A,,k)2
k 0

4.6 Asymptotic Normality Implies Asymptotic Uniformity

With the results of the above two sections, in particular Lemmas 10 and 11, we can

state the main result of this chapter.

Theorem 42. Let the combinatorial distribution a,,k have generating j'. *'i;,iil,.:.l* with

real, non-positive roots only. Then, i- /,, i/,/':.: noi,,n' l.:,i implies '- ;,, /,/l/:. unifo ,iiil.'

Proof. We retain the notation of Section 4.5. Assuming .ii-~!,i1 normality, Lemma 11

tells us

S(1+ oE)2 as n --oo. (48)
Y 1(1 + A,k )2
By Lemma 10, it suffices to show that this divergence implies

) 0 (4 9)

as n -- oo for all q and all 0 < r < q, where ( =exp If (r = -1 is a root of p,(x), then
q
p,((r) 0, so we can assume here that this is not the case in order to avoid pathologies.

Taking logarithms, (4-9) is equivalent to the condition

S(log (1 + A,k) log + n,k) 00. (4-10)
k=1
By Lemma 12 below, there exists a constant a > 0 which depends only on (r such that for

all n and k we have

og(+ ) log (1 + ,)-,k > a (Ank 2'
(1+ An,k)

Thus,
5 (log (1 + A,k) log 1r + A.|) > a ,'
k)k)> (1+A,k)2'
k=1 k 1
and (4-8) implies (4-10) which is itself equivalent to (4-9). E

Lemma 12. Let z = e0 for some 0 < 0 < 27r. Then, there exists a constant a = a(z) such

that for all x > 0,

f(x) log(1 + x)- loglz + x > a 2
(x + 1)2
Proof. Write f(x) as

f(x) +1 dt fx+z dt +1 dt
t J t JIx+z t

The length of the domain of integration is the function d(x) = x + 1 Ix + z and we have

d(x)
f(x) >
x+l

So, we are interested in a constant a such that

d(x) > a
x+l

for all x > 0. We first show that there is a constant a1 such that d(x) > a1 for all x > 1.

We have the formula Ix + z = (x2 + 2x cos 0 + 1)1/2. Therefore, for all x > 1,

d'(x) ( x + 1 (x2 + 2xcos0 + 1)1/2)
dx
2x + 2 cos 0
=1-
2(2 + 2x cos 0 + 1)1/2
x X2 + 2X Ccos 0 + cos2 0)1/2
1- (x2 + 2xcos 0 + 1

> 0.

Therefore, d(x) is monotonically increasing for positive x, and for all x > 1 it is greater

than d(1), which we denote by al.

Next we show there exist constants a2 and 6 such that d(x) > a2 for all 0 < x < S.

We have d'(0) = 1 cos 0, so there exists S > 0 such that for all 0 < x < 6

1 cos 0
d(x) > x
2

Moreover, as 1 < 1, for all 0 < x < 6,
X+1

1 cos 0 x
d(x) >
2 x+l'

so we let a2 = 1Co. Now we have constants for x E [0, j) U [1, oc). As [6, 1] is a compact

set, there exists a constant a3 such that d(x) > a3 -- for all x C [6, 1]. Finally, set =

min {ai, a2, a3}.

Example 43. The signless Stirling numbers of the first kind c,,k count n-permutations

with k 1. For each n, we have the generating pr ,;''."".:,/

n
Cn,() = nkXk (. + 1) ... + X n 1)
k= 1

So, for each n, C,(x) has the roots 0,-, -2,-. -n + We have the following limit.

lim j
noo (1 +j)2 +
j-1 j>_
j=1j
1 1
2 j
j>l1

S00

Therefore, by Lemma 11 and Theorem 42, the signless Stirling numbers of the first

kind are i-,i,;;;,/I. // ll'/ normal and thus ;,,.'.rm.

CHAPTER 5
ON THE ROOTS OF THE BELL POLYNOMIALS

5.1 Stirling Numbers of the Second Kind

This chapter is concerned with the generating polynomials for a classic combinatorial

distribution, the Stirling numbers of the second kind. Recalling the notation [n]

{1, 2,..., n}, we begin with the definition of a fundamental combinatorial object.
Definition 26. For n E P, a set partition of [n], or an n-set partition, is a set of

disjoint, nor., ijplI subsets of [n] called blocks whose union is [n].

Example 44. A set partition of [6] into three blocks is {{1, 4, }, {2}, {3, 5, 6}}.

The natural question posed by an enumerologist is how many such objects are there.

In particular, given non-negative integers n and k, how many set partitions are there of [n]

with k blocks? The answer is the Stirling number of the second kind S,n,k. By convention

we set So,o = 1 and Sn,k = 0 for all k > n > 0. The results of this chapter rest primarily on

the following recurrence relation.

Lemma 13. For all n > 0 and 1 < k < n + 1,

Sn+1,k = Sn,k-1 + kS,,k. (5 1)

Proof. Let r be a set partition counted by Sn+1,k. If {n + 1} is a block in r, we may

remove it, leaving a set partition of [n] with k 1 blocks, an object counted by Sn,k-1.

On the other hand, if there are other elements in that block of r which contains n + 1,

we simply remove n + 1 from that block and still have k blocks of n elements, an object

counted by Sn,k. This is clearly an injection and thus shows that the left hand side above

is less than or equal to the right hand side. Now, let 7 be a set partition counted by the

right hand side above. If 7 has k- 1 blocks, we append a block containing only n+ 1. Else,

we choose one of the k blocks in 7 into which we insert n + 1. Again we have an injection,

and the equality holds. O

Lemma 14. For all n > 1 and 1 < k < n, we have the exact formula

Sk i-) (k (5-2)
i=0
Furthermore, we have the i-;;l,,/i.i/'.: formulae

Sn,k k- 0((k 1)") (5-3)

k"
and

Sn,k k. (5-4)

Proof. A set partition of an n-set is constructed by placing the elements into k disjoint

sets. Consider instead the case where there are k labeled boxes and n labeled balls which

we place in the boxes such that no box is empty. Call this an (n, k)-placement and let an,k

be the number of (n, k)-placements. Then, an,k = k! Sn,k, as there are Sn,k Av-, ~ to place

the balls into unlabeled boxes, and k! v-i-, to label these boxes. So, it suffices to show that

the summation in (5-2) is an,k.

We apply the method of Inclusion-Exclusion. There are k" v--i- to distribute the

balls to the boxes, as we have k choices of what to do with each of n balls. However, we

may have overcounted distributions in which a box was left empty, so we subtract the

(k) (k 1)" V--I- to choose a box to be empty and distribute the balls to the remaining
k 1 boxes. Now, we have overcounted the distributions in which there are two empty

boxes. Indeed, for some 1 < p < q < k, consider the set of distributions which leave boxes

p and q empty. We counted such a distribution once in the first term, but subtracted

twice in the second term, once for leaving box p empty and once for leaving box q empty.

So, we need to add 1 for each such pair of boxes and each way to leave that pair empty

and distribute the balls to the other boxes, () (k 2)". This process continues until the

number i of empty boxes reaches zero, and we have the desired sum.

For the .ii-ii!ld lI ics, we write (5-2) as

k- k(k- t)- k(k- 1)(k- 2)-
Sk k! k! + 2k!

As n -+ oc, the dominant term is -. Furthermore, the dominant term of S,,k is

k a constant multiple of (k 1)", demonstrating (5 3). Finally, as n o0,

S. k O((k 1)")
k_ k1 '.
k! k!

Lemma 15. For all n > 1 and 1 < k < n, we have the bounds

k-k < S,k < (5-5)

and
k [k/2] -1 k k
(- 2i- 1)" < S,k < -. (5-6)
k! 2i1 k!
i 0
Proof. The upper bound, Sn,k < -L, follows trivially from (5-2). By the recurrence (5-1),

for all n > k > 1, we have

Sn,k > kSn-l,k

> k2S-_2,k

> k-kSk,k

= k-k

as Sk,k = 1 for all k, completing (5-5). The lower bound of (5-6) is the difference of the

dominant term and all odd-indexed, hence negative, terms of (5-2). D

5.2 Bell Polynomials

For all n > 1, let B,(x) be the generating polynomial for the Stirling numbers of the

second kind:

B, (x) Z= S,kXk.
k= 1
These are known as the Bell p y.i/;/n'; ,,i,.:.l

Example 45. The first few Bell i .1/;/,;,. iii,,.:, are as follows.

Bo(xr)= 1,

Bl(x) = x,

B2(x) =x + 2,

B3(x) = x + 32 + x3,

B4(x) x+7x2+6 3 +x4.

Lemma 16. The following recurrence relation on the Bell './;/ ,:'i "./l/ holds for all n > 0.

Bn+(x) = xB,(x) + xBj(x). (5-7)

Proof. Applying (5-1) to each term of B,+l(x), we have

Brz+l () Sn+l,kXk

= ((Sn,k-1 + kSf,k )Xk

S= .Sr,k-k + kSfl,kXk

= X Sn,k-1xk-I 1 + x kSn,kXk-1

xB,(x) + xB,(x).

We have the following well-known result on the roots of the Bell polynomials.

Lemma 17. For all n > 1, B(x) has n distinct, real, non-positive roots, including zero.

Proof. (Wilf, [40])

The statement holds for Bi(x). Assume by induction the statement holds for B,(x).

Multiplying each term in (5-7) by ex,

exB, 1(x) = x(exB,(x))'.

By Rolle's theorem, B,(x) has n 1 roots, one between each consecutive pair of roots

of B,(x). Multiplication by x guarantees a root at zero. Since exB,(x) approaches zero

as x -i -oo, its derivative has one more root to the left of the leftmost root of B,(x),

accounting for all n + 1 roots of B,+i(x). O

Thus we can factor each Bell polynomial as

B,(x) = x(x + A,,2) (x + n,n), (5-8)

where 0 = An,1 < An,2 < < An,, and n > 1. In this chapter we explore the roots

An,k, finding bounds and limits for each fixed k. First, we have a theorem on the Stirling

numbers which builds on the results of the previous chapter but was held aside for the

foregoing definitions and discussion.

Theorem 46. The Stirling numbers of the second kind Sn,k are ir;,,,,' I.:/ ,ill/// u:./f.rm.

Proof. Harper [19] proved that the Stirling numbers of the second kind are ,-mptotically

normal. The above discussion shows that their generating polynomials have real,

non-positive roots only, so by Theorem 42 in the previous chapter, we achieve the desired

result. O

Now, let us consider the roots more closely.

Example 47. We solve for the roots of the first few Bell pi..1 ;,,;.. i,,.l

BI(x) = x,

B2(x) x(x + ),

-3+ -3-
B3(x) x(x + -)(X + 2
2 2

For higher n, the il'. 1.,,, becomes 'n,,. ". ;, as can be ,' l./7:.; seen in the other Bell

1 1;1,,'.i',,. :,1l given in Example 45.

As our work often deals with sums of products of reciprocals of these roots, for

notational convenience we will write 6n,k = for k = 2, 3,..., n, or simply 6k = if n is

clear from context.

Lemma 18. For all n > 1 and 2 < k < n 1, we have

Sn,k I An,i ... n,i, k

where the sum is taken over all (n-k)-tuples {il, i2, ... in-k} with 2 < il < ... < in-k < n.

Proof. Expanding the factorization in (5-8), an xk term is achieved by choosing x from k

of the terms and A,,i from the remaining n k terms. The sum of all such products is S,,k,

the coefficient of xk in B.(x). E

We note that, for all n > 1, we have S,, = 1. Lemma 18 could be rewritten to reflect

this by allowing the sum over all (n n)-tuples to be 1.

Lemma 19. For all n > 2 and all no. m1i1i;I subsets M C {2,..., n},

__ ATn,i 6n,i,
iEM iEMC

where Mc = {2 < i < n| i f M} is the complement of M.

Proof. We first note that the product of the n 1 non-zero An,k's is S,i = 1, so

>2 >2
k>2 k>2

Thus, for any M C {2,..., n},

Sn,i ,J i = 1, SO
iEMC iEM
( )1

iEM iEMC

j i A. j 6n,j.
iEM iEMc

D

Lemma 20. For all n > 1 and 2 < k < n 1, we have

Sn,k >1 n,i .. ,i1, (5-9)

where the sum is taken over all (k 1)-tuples {ii, i2, ..., ik-} with 2 < il < ... < ik- < n.

Proof. Apply Lemma 19 to each (n k)-tuple in Lemma 18. D

For more on Stirling numbers of the second kind and Bell polynomials, particularly

their properties and myriad applications, the reader is referred to the classic by Riordan

[28] as well as the excellent texts by Roman [29] and Wilf [40].

5.3 Bounds on the Roots of the Bell Polynomials

Lemma 21. For all n > 2,
2"-1
--- < 6n,2 < 2n-1.
n
Proof. By Lemmas 14 and 20 and the nonnegativity of the 6,ji's,

n
K6,2< ,k Sn,2 < 2"-
k=2
As 6n,2 is the largest term of the sum, it is at least as large as the mean, so 6n,2 > 2

Finally, this last expression is at least 2-- for all n > 2.
no l n>2

Lemma 22. For all n > 3,

3n-1 4
2T < 6n,3 <
2" 9n2

3n-1
2 n

Proof. We begin with the upper bound. By Lemmas 14 and 20,

2 25i

1 -
Sn,3 23n-
2

3n-1
0(2-) < 2
2

(5-10)

In particular,

an-1
6n,2n,3 < 3`
2

Applying the inequality 6n,2 > T from Lemma 21,

3n-1
6n,3 < -
26n,2
n3 1
2"

For the lower bound, inequality (5-5) of Lemma 15 provides

C q-3
Sn,3 > -3

The next inequality follows from Lemma 20, the fact that the largest of the (2 1) terms in

(5-10) is 6n,2n,3, and our upper bound 6n,2 < 2-1.

Finally, dividing by 2"-1

3n-3
2n- 6n,3 > 3

3n-3
6n,3 >

3"-1
2"

2 3 -3
n(n 1)

n(n 1)
4
9n2"

Theorem 48. For each k > 2, there exist j.. I;;./.',,..l,.: fk(x) and Fk(x) of degrees 2k-2 and

2k-2 ,, t1, / :/; such that for all n > k

( k >" 1 < (k "Fk
k-1 fk(u) k-1

Proof. We induct on k, generalizing the proof of Lemma 22. We proved the statement for

6n,2 and 6n,3 in Lemmas 21 and 22, respectively. Let k > 4 and assume by induction that
the statement holds for all 6T,j with 2 < j < k. By Lemmas 14 and 20, for all n > k,

l -.i 6-,i4-1 = Sn,k < k! (5-11)

where the sum is taken over all (k-1)-tuples {il,,..., ***k- with 2 < ii < ... < ik1_ < n.

By induction we have the following lower bound for each 2 < j < k.

S j 1 f
,i2 (i )" fj(>

which holds for all n > k and where fj(x) is a polynomial of degree 2j-2 for each j

(independent of n). Define the degree 2j-1- 1 polynomial F(x) = -f2(x)f3(x) fk x).

Then,

k"
n,26n,3 n,k k
k"
6n,k <
k!,n,2 2 6n,k-1
k (k 2)T (k 3)T 2T
1 (k k 2)" 12-
F (n) k
(k 1)"

For the lower bound, inequality (5-5) of Lemma 15 provides

Sn,k > kn-k
Sn,k > kn

The largest of the (Q- ) terms in (5-11) is n,26n,3 ... 6n,k. By our induction

hypothesis, 6,,j obeys the following bound for each 2 < j < k.

6n,< F(n)-1

for polynomials Fj(x). Altogether,

]kn-k
6n,26n,3 6n,k > n-1
k-)

k n-k (k 2)n (k 3)n 2n 1
1 (k 1)n (k 2)n t fk(n)

where fk(x) (D)F2(x)F3(x) ... k- (x).

5.4 Asymptotics of the Roots of the Bell Polynomials

The next two lemmas use our bounds on 6,,2 and 6,,3 to obtain the .,i-mptotic

equivalences 6n,2 ~ 2"-1 and 6n,3 2n-

Lemma 23. We have the following limit.

lim = 1.
n-oo 2n-1

Proof. The inequality (n,2 < 2'-1 from Lemma 21 implies n, < 1 for all n > 2, and it

suffices to show that the terms 6 are bounded below by a sequence converging to 1. By

Lemmas 14 and 20 we know that S,,2 = 2n-1 1 is the sum of I,2, 6Tn,3, and n 3 other

terms, each of which is less than 6J,3, so for all n > 3 we have the following inequalities

(applying the bound n ()" > 6n,3)

6n,2 + n,3 > 2n-1

6n,2 + (2 1 > 2"-1,
(3\n
6n,2 > 2T 2n2 (
2
6n,2 2 (
2n- >1 4

The final limit, in essence 2n2 (I)T -- 0 as n -- oo, holds as 2n2 is a polynomial and

(Q)" is a geometric progression with positive ratio less than one.

Lemma 24. We have the following limit.

2"
lim n,3- 1 1
n->oo ,n-

Proof. Lemma 20 expresses Sn,3 as the sum of all products 6i6j. We can split this into two

sums, one sum containing those pairs which include 6n,2 and one sum containing all other

pairs,
3n-1
6n,2 6n,i + 6n,i6n,j = S,3 2 -(2"). (5-12)
i>3 3 We proceed by showing that the second summation above is negligible with respect to

the dominant term 3"-1. Lemma 22 gives us the upper bound 6,,3 < n -, and by our

ordering on the 6's we have 6,,j < 6n,3 for all j > 3, implying 6n,i6n,j < (.n" 2 for each of

the (",2) pairs i,j with 3 < i < j < n. Thus,

n 2 n3"_1 2 9 n4
S iK <( 2 2 ( j2 (5-13)
3
and

3 0 i 3

Combining (5-12) and (5-13), we have

3n-1 "
2 4

Dividing by our .-i~iil i..c approximation of 6 ,2 67n,2 ~ 2"-1

,i 2--- O Q\ n4

i>3

By Theorem 48, (I)F 4(n) > 6,4 > n,5 > ... for the fixed polynomial of degree 3 F4(x),

and

so

n-1
6n,3-31 -0((4) "l)

Finally, we divide by 3- and take the limit as n oo.

2n t 0 8) n T\t
;-- ((| n4)
6r,3 3-1 O 4
liM rn,3 2 lim 1 (( n4
lim 3- n4--
n*oo 3fl 1

= 1.

The following theorem generalizes these results to all k, giving the ..i-~''..i-ll ic

approximation 6n,k (k-1 for each fixed k.

Theorem 49. For each fixed k, we have the following limit.

(k 1)"
lim 6n,k ( 1.-.
n-oo k- 1

Proof. We generalize the proof of the previous lemma to all k > 4. We induct on k, the

cases k = 2 and k = 3 given in the previous two lemmas. The induction hypothesis gives

us .i',iii| ll Iic bounds which improve upon those given in the previous section. Indeed, for

all e > 0 there exists N > 0 such that n > N implies that for all 2 < j < k, we have

( j 1)
,; ( 1 (1 ,1 + ).

We can rewrite equation (5-9) of Lemma 20 as

k"
6n,26n,3 J 6n,k-1 > Jn,j + ^ 6n,j_, = Sn,k = k! O((k 1)"), (5-14)
j>k

where the second sum is taken over all (k 1)-tuples 2 < ji < .. < jk-1 < n containing

at least two elements which are each at least k. We first show that this summation is

negligible with respect to k". Let 2 < ji < .. < jk-1 < n be such a (k 1)-tuple. Then,

the product of the greatest k 3 6,,j's is bounded above by the product 6,n ,2 6n,k-2, and

the product of the remaining two J6,j's is bounded above by 52,k*

n,ji n,jk- < 6n,26n,3 ... n,k-26n,k6n,k
3n (k 2)T k" k"(
< 2" F3 (n) Fk-2(n) Fk(n)Fk(n)
2T (k 3)n (k 1)- (k 1)-
2 (k 2) (
(k (k1--) )2

where G(x) = F3x) Fk-2 x)Fk2(x). Set a =- k- and note that k 1 < a < k. The

summation was over a subset of all (k 1)-tuples, so we overcount the terms by setting

H(x) = G(x) () and we can therefore rewrite equation (5-14) as

k"
6 ,26n,3" 6n,k-, 6 ,j= O(a H(n)). (5-15)
j>k

Now, by induction we have

n,3" 1 (k 1)n-1
2n (k 2)-
(k 1)"-1
(k- 2)!

While the above was an .i-!,ii1ll ic approximation, we have the useful information

n,26n,3 ". 6n,k- 1

o ((k l)").

As a > k 1, this implies

6n,26n,3 6.n,k-1

O(c").

Dividing equation (5-15) by (k1),
(k-2)!

j>k

k"-1
(k 1)"

o (p' H( n)),

(516)

where k= (k 2) Now it suffices to annihilate all terms on the left hand side

except 6,,k. By Theorem 48,

(k Fk+(n) > 6n,k+l > .
\ k I

k"-1
kn 1
k (k 1)

k"-1
(k 1)"

i +l
i>k+1

0 (P"(H(n) + nFk+1(n))).

Taking the limit as n -- oo, we achieve our desired result.

O(pH((n))

REFERENCES

[1] N. Alon and J. Spencer. The Probabilistic Method. Wiley-Interscience, New York, NY,
2000.

[2] R. Arratia. On the Stanley-Wilf conjecture for the number of permutations avoiding a
given pattern. Electronic Journal of Combinatorics, 6(1):N1, 1999.

[3] E. A. Bender. Central and local limit theorems applied to .i-vmptotic enumeration.
Journal of Combinatorial Th(..,;, Ser. A, 15:91-111, 1973.

[4] M. B6na. Permutations avoiding certain patterns; the case of length 4 and
generalizations. Discrete Mathematics, 175:55 67, 1997.

[5] M. B6na. Combinatorics of Permutations. C'!h 11i, i1, & Hall, Boca Raton, FL, 2004.

[6] M. B6na. The limit of a Stanley-Wilf sequence is not ahv--, rational and 1 ,li 1-,
patterns beat monotone patterns. Journal of Combinatorial Th(..,It 110:223-235,
2005.

[7] M. B6na. On a balanced property of derangements. Electronic Journal of Combina-
torics, 13, 2006.

[8] M. B6na. A Walk Th,,..;,i, Combinatorics. World Scientific, River Edge, NJ, 2006.

[9] M. B6na. On a balanced property of compositions. Online Journal of A,.,rl;,.:.
Combinatorics, 2, 2007.

[10] M. Bousquet-Milou. Four classes of pattern-avoiding permutations under one roof:
Generating trees with two labels. Electronic Journal of Combinatorics, 9(2):R19,
2003.

[11] C. A. C'!ii 1 I oilides. Combinatorial Methods in Discrete Distributions.
Wiley-Interscience, Hoboken, NJ, 2005.

[12] F.R.K. Chuiin, R.L. Graham, V.E. Hoggatt Jr, and M. Kleiman. The number of
Baxter permutations. Journal of Combinatorial Theory (Series A), 24:382-394, 1978.

[13] K. L. Chuing. A Course In Pi,..,,:7Ii,. Th(..<,; Elsevier, San Diego, CA, 2001.

[14] M. Coleman, M. Albert, I. Leader, and R. Flynn. Permutations containing many
patterns. Annals of Combinatorics, to appear.

[15] Micah Coleman. An answer to a question by Wilf on packing distinct patterns in a
permutation. Electron. J. Combin., ll(1):Note 8, 4 pp. (electronic), 2004.

[16] H. Eriksson, K. Eriksson, S. Linusson, and J. Wastlund. Dense packing of patterns
in a permutation, Proceedings of the 15th Conference on Formal Power Series and
Algebraic Combinatorics (\1. !l>ourne, Australia). http://www.i3s.unice.fr/fpsac/
FPSAC02/articles.html, 2002.

[17] M. Fekete. Uber die Verteilung der Wurzeln bei gewissen algebraischen gleichungen
mit ganzzahligen koeffizienten. Mathematische Zeitschrift, 17:228 -249, 1923.

[18] P. Halmos. Measure The.. ;. D. Van N..-iI i.1I Co., Berlin, 1956.

[19] L. H. Harper. Stirling behavior is .,-i-~ !',ltically normal. Ann. Math. Statist.,
38:410-414, 1967.

[20] P. A. Hasto. The packing density of other 11,i,. -I permutations. Electronic Journal of
Combinatorics, 9(2):R1, 2002.

[21] M. Klazar. The Filredi-Hajnal conjecture implies the Stanley-Wilf conjecture. In
Formal Power Series and Al/, /,i.:.. Combinatorics, pages 250-255, Berlin, Germany,
2000. Springer Verlag.

[22] M. Klazar. Personal communication, 2008.

[23] A. Marcus and G. Tardos. Excluded permutation matrices and the Stanley-Wilf
conjecture. Journal of Combinatorial Theory Series A, 107:153-160, July 2004.

[24] D. Marinov and R. Radoicid. Counting 1324-avoiding permutations. Electronic
Journal of Combinatorics, 9(2):R13, 2002.

[25] Alison Miller. Asymptotic bounds for permutations containing many different
patterns, 2006. preprint.

[26] A. M. Odlyzko. Handbook of Combinatorics, volume 2, chapter Asymptotic
Enumeration Methods, pages 1063-1229. Elsevier, Cambridge, MA, 1995.

[27] A. Price. Packing Densities of L'n r Patterns. PhD thesis, University of
Pennsylvania, 1997.

[28] J. Riordan. An Introduction to Combinatorial A,.l,i-!. Wiley, New York, NY, 1980.

[29] S. Roman. The Umbral Calculus. Dover, Mineola, NY, 1984.

[30] A. Rucifiski. Random Graphs, chapter 2, Proving Normality in Combinatorics, pages
215-231. Wiley Interscience, Cambridge, UK, 1992.

[31] R. Simion and F. W. Schmidt. Restricted permutations. European Journal of
Combinatorics, 6:383 -406, 1985.

[32] N.J.A. Sloane. The on-line encyclopedia of integer sequences. http://www. research.
att.com/-njas/sequences/A088532, 2003.

[33] R. Stanley. Enumerative Combinatorics, Volume 1. Cambridge University Press,
Cambridge, UK, 1997.

[34] R. Stanley. Enumerative Combinatorics, Volume 2. Cambridge University Press,
Cambridge, UK, 1999.

[35] J. Taylor. An Introduction to Measure and Pi, ..',,7.~li Springer, New York, NY,
1997.

[36] V. Vatter. Permutations avoiding two patterns of length three. Electronic Journal of
Combinatorics, 9(2):R6, 2003.

[37] V. Vatter. Small permutation classes. http://arxiv.org/abs/0712.4006, 2007.

[38] D. Warren. O'l.:I, ..:,,,j the Packing Behavior of L.,. ., Permutation Patterns. PhD
thesis, University of Florida, 2005.

[39] J. West. Generating trees and forbidden subsequences. Discrete Mathematics, 157(1 -
3):363 374, 1996.

[40] H. Wilf. Generril.ilf, iI.'...A...,i~ ; A K Peters, Wellesley, MA, 2006.

[41] D. Zeilberger. A loving rendition of the Marcus-Tardos amazing proof of the
Fiiredi-Hajnal conjecture. http://www. math. rutgers. edu/~zeilberg/mamarim/
mamarimPDF/martar .pdf, 2003.

BIOGRAPHICAL SKETCH

A native Florida Cracker, I graduated from Seabreeze High School in 1994 and

entered college as a music 1 i i' r. Realizing my lack of the skill and devotion necessary

for a professional musician, I enlisted in the United States Navy, stationed for four years

in Yokosuka, Japan, where I met my wife Hiroko Shinohara. After my active duty period,

I completed an Associate's degree at Daytona Beach Community College and joined

my brother and his wife here in Gainesville, entering the world of higher math for the

first time at the University of Florida. As a junior I first met my advisor and fell in love

with Combinatorics. I graduated summa cum laude in 2004 and accepted a fellowship at

UF. I have thoroughly enjoi, d my graduate career, even my eight-month sabbatical to

Baghdad, Iraq last year, fully funded by the N i- I1 Reserve. I look forward to a long career

of mathematics research and hope to make some small contribution to the Conversation.

PAGE 1

1

PAGE 2

2

PAGE 3

3

PAGE 4

PAGE 5

page ACKNOWLEDGMENTS ................................. 4 LISTOFFIGURES .................................... 7 ABSTRACT ........................................ 8 CHAPTER 1INTRODUCTION .................................. 10 1.1AsymptoticEnumeration ............................ 10 1.2NotationforAsymptoticGrowthRates .................... 10 1.3GeneratingFunctions .............................. 11 2PATTERNAVOIDANCEINPERMUTATIONSAVOIDINGAMONOTONEPATTERN ...................................... 12 2.1PermutationsandPermutationPatterns ................... 12 2.2AnOpenProblembyM.Atkinson ...................... 24 2.3GeneratingTrees ................................ 26 2.4\Hat"Notation ................................. 31 2.5MonotoneIncreasingPatternsq 33 2.6ThePatternq=123 .............................. 38 3PATTERNPACKING ................................ 53 3.1GeneralPatternPacking ............................ 53 3.2PatternPackingin123-avoidingPermutations ................ 57 3.3PatternPackinginq-avoidingPermutations ................. 59 3.4PackingDensityandFurtherDirections .................... 61 4ASYMPTOTICNORMALITYANDUNIFORMITY ............... 63 4.1ProbabilityTheory ............................... 63 4.2TriangularArrays ................................ 64 4.3AsymptoticNormality ............................. 65 4.4AsymptoticUniformity ............................. 66 4.5GeneratingPolynomialswithReal,Non-PositiveRoots ........... 68 4.6AsymptoticNormalityImpliesAsymptoticUniformity ........... 71 5ONTHEROOTSOFTHEBELLPOLYNOMIALS ................ 74 5.1StirlingNumbersoftheSecondKind ..................... 74 5.2BellPolynomials ................................ 77 5.3BoundsontheRootsoftheBellPolynomials ................. 80 5.4AsymptoticsoftheRootsoftheBellPolynomials .............. 83 5

PAGE 6

....................................... 88 BIOGRAPHICALSKETCH ................................ 91 6

PAGE 7

Figure page 2-1Thepermutation3142. ................................ 13 2-2Thepermutation532614. ............................... 25 2-3Thepermutation865321947. ............................. 27 2-4Arootedtree. ..................................... 28 2-5Thecompletebinarytree. .............................. 29 2-6TheFibonaccitree. .................................. 30 2-7T(123,132). ...................................... 32 2-8TreeinW(123,231)rootedat42153 ......................... 40 2-9Thelayeredpermutation213654withlayers21,3,and654 ............ 41 3-1ThepermutationW(6)=342516 ........................... 55 7

PAGE 8

8

PAGE 9

9

PAGE 10

26 ]intheHandbookofCombinatorics.

PAGE 11

40 ],Stanley[ 33 ],[ 34 ],andBona[ 8 ]. 11

PAGE 13

2-1 Thepermutation3142. Inallcontextsconsideredhere,apermutationpatternorsimplyapatternisitselfapermutationbuttherearesubtledierenceswhichweshallexploit.Givenapermutation=1n,asubsequenceofisanorderedsubsetoftheentriesof,(i1;:::;ik)forsomek,whichwewriteinthesameorderastheyappearin,soi1
PAGE 15

3 byreducingallsubsequences,inthesameorderingasabove:Length0:;Length1:11111Length2:12121212122112211212Length3:123132123132123123231123213213Length4:13421234132413242314Length5:13425Ofcourse,wecouldcontinueinthisfashion.Itisagreatexerciseforthebeginnerinthisareatoexhaustthesubsequencesofapermutationtodeterminewhatpatternsthepermutationcontainsoravoidsandposesomeconjectures.Thisishowonelearnsanythingincombinatorics,by\gettingourhandsdirty",doingenoughmanuallaboronourcombinatorialobjectstogetafeelfortheirgrowthandotherproperties. 15

PAGE 16

PAGE 17

PAGE 18

18

PAGE 19

8 ,thisisequivalenttothestatementthat,foreachpairiRnpj+1ifandonlyifRni+1>Rni+1:AswerunthroughallpairsiRpjifandonlyifRi>Rj:Altogether,containingisequivalenttoRcontainingR.Likewise,containingisequivalenttoCcontainingCand1containing1.Ofcourse,wecanreplacetheword\containing"bytheword\avoiding"ineachstatement.Critically,asthethreemapsR;C;and1arebijectionsSn!Sn,itfollowsthatthenumberofn-permutationswhichavoidisthesameasthenumberofn-permutationsavoidingR;C;1.Wehaveachievedthefollowingresult.

PAGE 20

31 ])Foralln0,sn(123)=sn(132): 20

PAGE 21

So,infactweseethatforall;2S3andn0;sn()=sn():Onemaybetemptedtosuspectsuchastatementholdsforpatternsofeverylengthm.However,withacomputercheckorafewpagesofscribbling,oneobtainss6(1342)=5126=s6(1234)=513:Wenowturntoasymptotics.In1980,RichardStanleyandHerbWilfindependentlyconjecturedthatforeachpatternthereexistsaconstantcsuchthat,foralln0,wehavesn()cn:In[ 2 ],Arratiaprovedthatthiswasequivalenttothefollowing,longknownastheStanley-WilfConjecture. 23 ]oftheFuredi-HajnalConjectureonpermutationmatrices.ThattheFuredi-HajnalConjectureimpliestheStanley-WilfconjecturewasprovenbyKlazar 21

PAGE 22

PAGE 23

PAGE 24

3 ,thelimitandlimsupagree,andLisaswewantittobe. 2-2 .Notethatinourpreviousexampleeachentryofthebackendislessthanthethreshold.Creatingourowngoodluck,wechose532614forourexamplespecicallybecauseitavoids123.Infact,every123-avoidingpermutationsharesthisproperty,asimplestructureofwhichweshalltakegreatadvantageinourhandlingofthesepermutations.Toseethisproperty,supposeourthresholdistandthereisanentryjinthebackend(equivalentlyj>t)withj>t.Bydenitionofascendee,t1
PAGE 25

Thepermutation532614.

PAGE 26

PAGE 27

PAGE 28

QQQQQQQQQ Arootedtree. canbegeneratedfromanyn-objectisallweneedtoknow,sowemightaswelllabeleachnodewithitsdepth.In[ 39 ]Westbeginswithatrivialexample,thecompletebinarytree.Webeginwitharootwithlabel(2).Oursuccessionruleisthateachnodewithlabel(2)hastwochildrenalsolabeled(2). 39 ],Example1). 28

PAGE 29

Thecompletebinarytree. 39 ],Example3).

PAGE 30

AAAAAAAAqqqqqqqqqFigure2-6. TheFibonaccitree. Foradetailedexpositionontheuseofgeneratingtreesinthestudyofpatternavoidance,see[ 39 ],[ 12 ],[ 36 ],[ 10 ]and[ 24 ].Herewedenethegeneratingtreeswhichwillbeusedthroughout.Thesedenitionsdependonthepatternsandqwhicharebeingavoided,soweassumethepatternstobegiven.Thiswillbeclearfromcontext.Firstweexplainthemotivations.Recallour0notation.Forapattern2Sm,thepattern02Sm+1isobtainedbyprependingwiththeentry(m+1).Thefundamentalquestionhereiswhethervariouslimits(orlimitsuprema)forthenumberofpermutationswhichavoidsomepatternarethesameasthosewhichavoid0(assumingfornowthatthelimitsexist).Itwasnotedabovethat-avoidanceimplies0-avoidance,butthereare0avoiderswhichcontain.So,ourquestionboilsdowntojusthowmanyofthesethereare,inparticularwhataretheasymptoticsofthesepermutationswithrespecttothesetof-avoiders.Wewould 30

PAGE 31

2-7 .Anactivesiteinapermutationisavalidinsertionpoint,thatis,asitewherewecaninsertn+1andobtainachildwhichisstillinthecurrentgeneratingtree,soforourpurposesanactivesiteissuchthattheinsertionwillnotcauseanoccurenceofanypatternwhichweseektoavoid.Thedepthofapermutationisthenumberofactivesitesin,equivalenttothenotionofdepthdenedaboveongeneratingtrees.Wenotethatthedepthdependsonboththepermutationitselfandonthetree,specicallythepatternbeingavoidedwhichdeterminesthetree. 31

PAGE 32

4231(2)@@@@@@@@213(1) 4213(2)eeeeeeee12(2)BBBBBBBB312(2)4312(3)BBBBBBBB3412(1)qqqqqqqqqqqqqqqqqqqqqqqqFigure2-7. T(123,132). to2=4,and^3refersto4=3.Ontheotherhand,thepermutation1432containsthree^'s,namely143,142,and132.Inthiscasewecanrefertothe^1,theentry1,butwehaveseveral^2'sandseveral^3's.Itshouldalsobenotedthatoneentrycouldbebotha^ianda^jforsomei6=j.

PAGE 33

4 ].

PAGE 34

Proof. Proof. 34

PAGE 35

2 thenumberofsuchweakn-classesisatmostapolynomialfunctionf(n).Thereforeourovercountofn-permutationsisf(n)sn1(q;).Wearenowinpositiontotakeourlimits.L(q;0)=limsupn!1sn(q;0)1=nlimsupn!1(f(n)sn1(q;))1=n=limsupn!1f(n)1=nsn1(q;)1=n=limsupn!11sn1(q;)1=n=L(q;):CombinedwiththeknowledgethatL(q;0)L(q;),wearenished. Thefollowinglemmafrom[ 4 ]and[ 5 ]providesanupperboundonthenumberofpermutationsoflengthnwhichavoidtheincreasingpatternoflengthr. 35

PAGE 36

Proof. Withthislemmainhand,wesubtlyalteranotherproofofBonatoachieve: 6 thereareatmost(r1)2(k1)possiblepermutations 36

PAGE 37

Inparticular,forq=123,wehave(r2)2=1,sowiththeassumptionsn(123;)cnforalln,wendsn(123;0)(c+1)n1foralln.TheStanley-WilfConjecture(Marcus-TardosTheorem)tellsusthatforanypatternorsetofpatternsthereissuchaconstantcasintheabovehypothesis.Inthecaseofavoidingasinglepattern,Arratiashowedin[ 2 ]thatthesequencesn()1=nisincreasing.However,therearesetsofpatternsforwhichthesequencesn()1=nisnotincreasing.Thus,takingctobetheleastconstantsuchthatsn()cnforallnN.Inparticular,theconstantcmaybesignicantlygreaterthanL(),sothenewconstantdisclosertoourlimitandthusabetterindicatoroftheasymptoticbehaviorofoursequencesn().Suchasituationmotivatesastrengtheningofthepreviousproposition. 37

PAGE 39

Proof. 2-8 39

PAGE 40

4762153qqq TreeinW(123,231)rootedat42153 Denotingbyaj;kthenumberofpermutationsatthejthlevelwithdepthk,0jand1kd,wehavetherecursivesystema0;d=1a0;k=08k6=daj;k=dXt=kaj1;t8j1;1kd:Fromthisrecursivesystemweseethataj;k=dk+1+jdk+1.Foreachdandjwemaysumoverallktoattainthelevel-numberd+j1d1.However,acombinatorialproofispreferable.Weknowthatd+j1d1countsj-multisetsof[d],andsuchamultisetwritteninnonincreasingorderspellsouttheorderofactivesiteschoseninthelineagefromtoapermutationoflengthn+j. 40

PAGE 41

5 ]Denition5.33)Apermutationiscalledlayeredifitcanbewrittenastheconcatenationq1q2qkwhereeachqiisadecreasingsequenceofconsecutiveintegersandtheleadingentryofqiissmallerthantheleadingentryofqi+1for1ik1. 2-9 Thelayeredpermutation213654withlayers21,3,and654 Manyresultsareknownconcerningpatternavoidanceandpatternpackingforlayeredpermutations.OnecanseeSection5.2.2of[ 5 ],[ 27 ],[ 6 ],and[ 20 ].Ournextresultisonlayeredpatternswithjusttwolayers,equivalentlynon-monotonelayeredpatternswhichavoid123. 41

PAGE 42

42

PAGE 43

23 .Forexample,foranyd>2,wehavethe(123,3142)-avoidingpermutation(d+1)(d1)(d2)1(d+2)d,whichhasdepthd.Thismeanswecanarenotguaranteedaxedboundtothedegreesofourgeneratingpolynomialsofthelevel-numbersofourtrees.Allisnotlost,however,aswecanshowthatforanyofaclassofpatternsandanykthereisanupperboundtothenumberoftreesinWwhoseroothasdepthwhichisk

PAGE 44

2!n1p 2!n!;fromwhichweseethatFnisasymptoticallyapproximatedbythenthpowerofthedominantterm,thegoldenratio,Fn1+p 2!n:Furthermore,asthegoldenratioenjoysthepropertyofbeingasolutiontotheequationx2=x+1,sothat1+p 22=1+p 2+1=3+p 2,weimmediatelyachieveanapproximationforeveryotherFibonaccinumberF2n+13+p 2!n:

PAGE 45

2: 45

PAGE 46

2followsimmediatelyfromtheabovediscussionoftheasymptoticsoftheFibonaccinumbers. 2: 46

PAGE 47

47

PAGE 48

48

PAGE 49

Proof. 49

PAGE 50

25 .Letd0denotethedepthof.NotethatanyrootinWcontainsinitsfrontendacopyofthefrontendofandthereforehasdepthwhichisatleastd0.Letdd0.Webeginwithaprototype,apermutationinSk(123;)forsomek,andinsertdd0entriesintoconstructarootinWwithdepthd.Let2Sk(123;)haveachild(0)inUwhichisarootinWandhasthesamedepthas,d0.So,thereexistsanindextsuchthatinsertingk+1attingivesus(0),andcontainsacopyofnm.Ifd=d0,wetake(0)tobeourrootinW.Otherwise,webuild(dd0)inductively.Letrbetheindexoftherightmost^1of(0),andnotethatbyconstructiontistheindexofthethresholdof(0),and12t1=(0)1(0)2(0)t1.Besidesthethreshold,wewillinserteachentryusingoneoftwomethods.Onemethodistoinsertatamaximaldecreasinginterval,i.e.chooseanindexinamaximaldecreasingintervalinthefrontendof(j)andinatethepermutationatthatindexbythepermutation21. 50

PAGE 51

9 ,weknowthatthefrontendof(0)hasatmostmmaximaldecreasingintervals.Infact,thereareexactlyMmaximaldecreasingintervalsinthesubsequencer+1t1forsomeM
PAGE 53

Inthepreviouschapterwediscussedequivalenceclassesforpatternavoidance.Ifthepermutationcontainsthepattern,then1contains1.SimilarlyforRandC.Fromthisweimmediatelyachieveforallpermutationspat()=pat(1)=pat(R)=pat(C):So,;1;R;andCareinthesameequivalenceclasswithrespecttothefunctionpat.Next,wedenethefunctionmaxpatoverallnonnegativeintegersbymaxpat(n)=maxfpat()j2Sng:

PAGE 54

3-1 foranexample.Theclaimisthatforalln1,pat(W(n))Fn,whereFnisthenthFibonaccinumber,asdenedintheprevioussection. 2liminfn!1maxpat(n)1=nlimsupn!1maxpat(n)1=n2:

PAGE 55

ThepermutationW(6)=342516

PAGE 56

2!n!:Theupperboundonthelimitsupremumistrivialformaxpat,whichisnecessarilyboundedabovebythenumberofsubsequencesofann-permutation,2n. Basedonempiricalevidence,(seesequenceA088532intheOn-LineEncyclopediaofIntegerSequences[ 32 ]),itseemedthisnumbermay\approach"thetrivialupperboundof2n(thenumberofallsubsetsofann-permutation).In[ 15 ]thisauthorconstructedaclassofpermutationsoverwhichitwasshownthatlimn!1maxpat(n)1=n=2:However,whileconrmingoursuspicionthatthenthrootapproaches2,thisresultstillleftopenthepossibilitythatmaxpat(n) 2n!0asn!1,i.e.thepossibilitythatmaxpat(n)=o(2n).RecentlyMiller[ 25 ]andAlbertetal.[ 14 ]independentlyprovedwitharenementoftheoriginalclassfrom[ 15 ]andmoredelicatecountingtechniquesthatindeedlimn!1maxpat(n) 2n=1;i:e:maxpat(n)2n:Inparticular,in[ 25 ],Millershowedthewonderfullyexactbounds2nOn22np

PAGE 57

Thesignicanceoftheabovelimitisthatonemayhopetopackjustasmanypatternsinto123-avoidingpermutationsasinthegeneralcase.Furthermore,foranyincreasingpatternq=12rwithr>3,thesameupperboundholds,albeittrivial. 34 givesushopethat,evenwiththe123-avoidingrestriction,wemaypackasmanypatternsaswewouldlikeinapermutationofsucientlength.Experimentationsupportssuchaconjecture.Herewegiveconstructionsforq-avoidingpermutationswith\many"patternsforincreasingpatternsq.Ourconstructionsaremodeledonthoseof[ 15 ],[ 16 ],[ 14 ],and[ 25 ]andmeetorsurpasstheoriginallowerbound 57

PAGE 58

2n.WenotethatWilf'soriginalconstructionis132-avoiding,sowealreadyhaveitestablishedthatmaxpat132(n)>1+p 2n.Webeginwithafamilyof123-avoidingpermutationswhichshowthatthelimitinmumofmaxpat123(n)1=nisatleast1+p 2asn!1.WedeneourpermutationsP(n)inductively.LetP(1)=1andP(2)=12.Foreachoddn3,letP(n)=nP(n1),i.e.P(n1)prependedwithn.Foreachevenn4,letP(n)beP(n1)withninsertedimmediatelyafter1.So,wehaveP(1)=1;P(2)=12;P(3)=312;P(4)=3142;P(5)=53142;P(6)=531642:So,eachP(n)consistsoftwodecreasingsubsequences,andthusavoids123.Thenextpropositiongivesalowerboundonpat(n)withaproofsimilartotheproofgivenforProposition 33 2liminfn!1maxpat123(n)1=nlimsupn!1maxpat123(n)1=n2:

PAGE 60

60

PAGE 61

Itmaybenotedthattheseclassesofpermutationsresembletheclassesusedin[ 15 ],[ 16 ],[ 14 ],and[ 25 ],exceptthatwhereasineachofthesepaperstheconstructionconsistedofkrowsofkentrieseach,orastrippeddownversionofthat,thepermutationsweareusingherehaverestrictedrowlengths(numberofmaximaldecreasingsubsequences)duetotheq-avoidingrestriction.Itwasnotedthatk!1.Fromthisfactweseethatasweletr!1,ourconstructionsforincreasingpatternsoflengthr+1\approach"theconstructionsforthegeneralcaseandtheliminfsapproach2. 61

PAGE 62

38 ]andAlkesPrice[ 27 ].HerbWilfhasrecentlysuggestedthatthenotionofpackingdensitybeexaminedintheq-avoidingenvironment.Whydoweraisetheissueofthesedualities?Wesawinthesectionofpatternavoidancethatintherestrictedenvironmentsof123-avoidingpermutations,ormoregenerallyq-avoidingpermutations,itismorelikelythatarandomlychosenpermutationavoidssomepattern,or,critically,thechanceofavoidingapproachesorequalsthechanceofavoiding0,astatementwhichisusuallyfalseinthegeneralcase.So,restrictingtoq-avoidingpermutationsmakesourlifeeasierinthatwork.However,patternpackingbecomesmoredicult.Indeed,ifweconsiderallapproachesusedtoprovepatternpackingorsuperpatterns,theytakeadvantageofalatticeorcheckerboardstructureintheclassofconstructionstoboundmaxpat.Ifwerestrictthelengthoftheincreasingsequencesallowed,welosethisstructure.So,themeta-questioniswhetherweloseourasymptoticlimitsorsimplyneedmoredelicatetechniquestoseethem. 62

PAGE 63

18 ],Taylor[ 35 ],andChung[ 13 ].Fora 63

PAGE 64

1 ]andCharalambides[ 11 ]. Denition22. 19 ].Incombinatorialapplicationsan;kcountsobjectsofsizenwithsomestatistick.Forexample,inthenextchapterwewillstudythenumbersSn;k,countingsetpartitionsofannelementsetintokblocks.Givensuchasequencean;k,wesetsn=an;1+an;2++an;mforeachnandconstructanewsequencebn;k=an;k 2 64

PAGE 65

35 ]. 65

PAGE 66

7 ],[ 9 ]).

PAGE 67

7 ]and[ 9 ] q,aprimitiveqthrootofunity. Proof. 4{5 )isageometricsum,andwetakeadvantageofthepropertythatthe(kr)-rootsofunityaddupto1toseethatq1Xt=0(kr)t=8>><>>:0ifkr6=1:qifkr=1:

PAGE 68

4{5 )forwhichqdoesnotdividekr,and( 4{2 )reducestoSn;r()=qm=qXj=1an;jq+r;or (4{6)Sn;r() 4{7 )asthefunctionfromthedenitionofq-balance.So,q-balanceisequivalenttothisexpressionconvergingto1 4{1 )wecandropourassumptionthatqdividesm,andwehavethelimitlimn!1Sn;r()

PAGE 69

19 ]andalsousedin[ 30 ].Byourassumptionontherootsofourgeneratingpolynomials,wecanfactortheprobabilitygeneratingpolynomialpn(x)aspn(x)=an;m 1+n;k:FromthisthemeanandvarianceofeachIn;kfollowimmediately.E(In;k)=1 1+n;k:VarIn;k=E(I2n;k)(E(In;k))2=1 1+n;k1 (1+n;k)2=n;k 1+n;kx:

PAGE 70

3 ]whichfollowsfromTheorem 39 : 41 andkeepingtheabovenotation,wehaveshownthefollowing.

PAGE 71

10 and 11 ,wecanstatethemainresultofthischapter. Proof. 4.5 .Assumingasymptoticnormality,Lemma 11 tellsus 10 ,itsucestoshowthatthisdivergenceimplies q.Ifr=1isarootofpn(x),thenpn(r)=0,sowecanassumeherethatthisisnotthecaseinordertoavoidpathologies.Takinglogarithms,( 4{9 )isequivalenttothecondition 12 below,thereexistsaconstant>0whichdependsonlyonrsuchthatforallnandkwehavelog(1+n;k)logjr+n;kj>n;k 4{8 )implies( 4{10 )whichisitselfequivalentto( 4{9 ). 71

PAGE 72

tZjx+zj1dt t=Zx+1jx+zjdt t:Thelengthofthedomainofintegrationisthefunctiond(x)=x+1jx+zj,andwehavef(x)>d(x) x+1forallx0.Werstshowthatthereisaconstant1suchthatd(x)>1forallx>1.Wehavetheformulajx+zj=(x2+2xcos+1)1=2.Therefore,forallx>1,d0(x)=d dx(x+1(x2+2xcos+1)1=2)=12x+2cos x2+2xcos+11=2>0:Therefore,d(x)ismonotonicallyincreasingforpositivex,andforallx>1itisgreaterthand(1),whichwedenoteby1.Nextweshowthereexistconstants2andsuchthatd(x)>2forall00suchthatforall0x1cos

PAGE 73

x+1;sowelet2=1cos x+1forallx2[;1].Finally,set=minf1;2;3g. 21 11 andTheorem 42 ,thesignlessStirlingnumbersoftherstkindareasymptoticallynormalandthusuniform.

PAGE 74

74

PAGE 75

5{2 )isan;k.WeapplythemethodofInclusion-Exclusion.Thereareknwaystodistributetheballstotheboxes,aswehavekchoicesofwhattodowitheachofnballs.However,wemayhaveovercounteddistributionsinwhichaboxwasleftempty,sowesubtractthek1(k1)nwaystochooseaboxtobeemptyanddistributetheballstotheremainingk1boxes.Now,wehaveovercountedthedistributionsinwhichtherearetwoemptyboxes.Indeed,forsome1p
PAGE 76

5{2 )asSn;k=kn 5{3 ).Finally,asn!1,Sn;k kn 5{2 ).Bytherecurrence( 5{1 ),foralln>k1,wehaveSn;kkSn1;kk2Sn2;kknkSk;k=knk;asSk;k=1forallk,completing( 5{5 ).Thelowerboundof( 5{6 )isthedierenceofthedominanttermandallodd-indexed,hencenegative,termsof( 5{2 ). 76

PAGE 77

5{1 )toeachtermofBn+1(x),wehaveBn+1(x)=XSn+1;kxk=X(Sn;k1+kSn;k)xk=XSn;k1xk+XkSn;kxk=xXSn;k1xk1+xXkSn;kxk1=xBn(x)+xB0n(x): 77

PAGE 78

Proof. 40 ])ThestatementholdsforB1(x).AssumebyinductionthestatementholdsforBn(x).Multiplyingeachtermin( 5{7 )byex,exBn+1(x)=x(exBn(x))0:ByRolle'stheorem,B0n(x)hasn1roots,onebetweeneachconsecutivepairofrootsofBn(x).Multiplicationbyxguaranteesarootatzero.SinceexBn(x)approacheszeroasx!,itsderivativehasonemoreroottotheleftoftheleftmostrootofBn(x),accountingforalln+1rootsofBn+1(x). ThuswecanfactoreachBellpolynomialas Proof. 19 ]provedthattheStirlingnumbersofthesecondkindareasymptoticallynormal.Theabovediscussionshowsthattheirgeneratingpolynomialshavereal,non-positiverootsonly,sobyTheorem 42 inthepreviouschapter,weachievethedesiredresult. Now,letusconsidertherootsmoreclosely. 78

PAGE 79

2)(x+3p 2):Forhighern,thealgebrabecomesunwieldy,ascanbereadilyseenintheotherBellpolynomialsgiveninExample 45 .Asourworkoftendealswithsumsofproductsofreciprocalsoftheseroots,fornotationalconveniencewewillwriten;k=1 Proof. 5{8 ),anxktermisachievedbychoosingxfromkofthetermsandn;ifromtheremainingnkterms.ThesumofallsuchproductsisSn;k,thecoecientofxkinBn(x). Wenotethat,foralln1,wehaveSn;n=1.Lemma 18 couldberewrittentoreectthisbyallowingthesumoverall(nn)-tuplestobe1. Proof.

PAGE 80

Proof. 19 toeach(nk)-tupleinLemma 18 FormoreonStirlingnumbersofthesecondkindandBellpolynomials,particularlytheirpropertiesandmyriadapplications,thereaderisreferredtotheclassicbyRiordan[ 28 ]aswellastheexcellenttextsbyRoman[ 29 ]andWilf[ 40 ]. Lemma21. 14 and 20 andthenonnegativityofthen;i's,n;2nXk=2n;k=Sn;2<2n1:Asn;2isthelargesttermofthesum,itisatleastaslargeasthemean,son;2>2n11 80

PAGE 81

9n2n;3<3n1 14 and 20 23n1O(2n)<3n1 21 ,n;3<3n1 5{5 )ofLemma 15 providesSn;3>3n3:ThenextinequalityfollowsfromLemma 20 ,thefactthatthelargestofthen12termsin( 5{10 )isn;2n;3,andourupperboundn;2<2n1.2n1n;3>3n3 9n2:

PAGE 82

k1n1 k1nFk(n): 22 .Weprovedthestatementforn;2andn;3inLemmas 21 and 22 ,respectively.Letk4andassumebyinductionthatthestatementholdsforalln;jwith2jknk:

PAGE 83

5{11 )isn;2n;3n;k.Byourinductionhypothesis,n;jobeysthefollowingboundforeach2jknk 21 impliesn;2 14 and 20 weknowthatSn;2=2n11isthesumofn;2;n;3,andn3otherterms,eachofwhichislessthann;3,soforalln>3wehavethefollowinginequalities 83

PAGE 84

2n>n;3)n;2+nn;3>2n1;n;2+n23 2n>2n1;n;2>2n12n23 2n;n;2 4n!1:Thenallimit,inessence2n23 4n!0asn!1,holdsas2n2isapolynomialand3 4nisageometricprogressionwithpositiveratiolessthanone. 20 expressesSn;3asthesumofallproductsij.Wecansplitthisintotwosums,onesumcontainingthosepairswhichincluden;2andonesumcontainingallotherpairs, 22 givesustheupperboundn;3
PAGE 85

5{12 )and( 5{13 ),wehaven;2Xi3n;i=Sn;3O9 4nn4=3n1 4nn4:Dividingbyourasymptoticapproximationofn;2,n;22n1,Xi3n;i=3n1 8nn4:ByTheorem 48 ,4 3nF4(n)>n;4>n;5>forthexedpolynomialofdegree3F4(x),and9 8nn4+(n3)4 3nF4(n)=O4 3nn4;son;3=3n1 3nn4:Finally,wedivideby3n1 9nn4:limn!1n;32n 9nn4=1: 85

PAGE 86

5{9 )ofLemma 20 as (k1)2nG(n);whereG(x)=F3(x)Fk2(x)F2k(x).Set=k2(k2) (k1)2andnotethatk1<
PAGE 87

5{15 )by(k1)n1 k1=k2(k2) (k1)3.Nowitsucestoannihilatealltermsonthelefthandsideexceptn;k.ByTheorem 48 ,k+1 87

PAGE 88

[1] N.AlonandJ.Spencer.TheProbabilisticMethod.Wiley-Interscience,NewYork,NY,2000. [2] R.Arratia.OntheStanley-Wilfconjectureforthenumberofpermutationsavoidingagivenpattern.ElectronicJournalofCombinatorics,6(1):N1,1999. [3] E.A.Bender.Centralandlocallimittheoremsappliedtoasymptoticenumeration.JournalofCombinatorialTheory,Ser.A,15:91{111,1973. [4] M.Bona.Permutationsavoidingcertainpatterns;thecaseoflength4andgeneralizations.DiscreteMathematics,175:55{67,1997. [5] M.Bona.CombinatoricsofPermutations.Chapman&Hall,BocaRaton,FL,2004. [6] M.Bona.ThelimitofaStanley-Wilfsequenceisnotalwaysrationalandlayeredpatternsbeatmonotonepatterns.JournalofCombinatorialTheory,110:223{235,2005. [7] M.Bona.Onabalancedpropertyofderangements.ElectronicJournalofCombina-torics,13,2006. [8] M.Bona.AWalkThroughCombinatorics.WorldScientic,RiverEdge,NJ,2006. [9] M.Bona.Onabalancedpropertyofcompositions.OnlineJournalofAnalyticCombinatorics,2,2007. [10] M.Bousquet-Melou.Fourclassesofpattern-avoidingpermutationsunderoneroof:Generatingtreeswithtwolabels.ElectronicJournalofCombinatorics,9(2):R19,2003. [11] C.A.Charalambides.CombinatorialMethodsinDiscreteDistributions.Wiley-Interscience,Hoboken,NJ,2005. [12] F.R.K.Chung,R.L.Graham,V.E.HoggattJr,andM.Kleiman.ThenumberofBaxterpermutations.JournalofCombinatorialTheory(SeriesA),24:382{394,1978. [13] K.L.Chung.ACourseInProbabilityTheory.Elsevier,SanDiego,CA,2001. [14] M.Coleman,M.Albert,I.Leader,andR.Flynn.Permutationscontainingmanypatterns.AnnalsofCombinatorics,toappear. [15] MicahColeman.AnanswertoaquestionbyWilfonpackingdistinctpatternsinapermutation.Electron.J.Combin.,11(1):Note8,4pp.(electronic),2004. [16] H.Eriksson,K.Eriksson,S.Linusson,andJ.Wastlund.Densepackingofpatternsinapermutation,Proceedingsofthe15thConferenceonFormalPowerSeriesandAlgebraicCombinatorics(Melbourne,Australia). 88

PAGE 89

[17] M.Fekete.UberdieVerteilungderWurzelnbeigewissenalgebraischengleichungenmitganzzahligenkoezienten.MathematischeZeitschrift,17:228{249,1923. [18] P.Halmos.MeasureTheory.D.VanNostrandCo.,Berlin,1956. [19] L.H.Harper.Stirlingbehaviorisasymptoticallynormal.Ann.Math.Statist.,38:410{414,1967. [20] P.A.Hasto.Thepackingdensityofotherlayeredpermutations.ElectronicJournalofCombinatorics,9(2):R1,2002. [21] M.Klazar.TheFuredi-HajnalconjectureimpliestheStanley-Wilfconjecture.InFormalPowerSeriesandAlgebraicCombinatorics,pages250{255,Berlin,Germany,2000.SpringerVerlag. [22] M.Klazar.Personalcommunication,2008. [23] A.MarcusandG.Tardos.ExcludedpermutationmatricesandtheStanley-Wilfconjecture.JournalofCombinatorialTheorySeriesA,107:153{160,July2004. [24] D.MarinovandR.Radoicic.Counting1324-avoidingpermutations.ElectronicJournalofCombinatorics,9(2):R13,2002. [25] AlisonMiller.Asymptoticboundsforpermutationscontainingmanydierentpatterns,2006.preprint. [26] A.M.Odlyzko.HandbookofCombinatorics,volume2,chapterAsymptoticEnumerationMethods,pages1063{1229.Elsevier,Cambridge,MA,1995. [27] A.Price.PackingDensitiesofLayeredPatterns.PhDthesis,UniversityofPennsylvania,1997. [28] J.Riordan.AnIntroductiontoCombinatorialAnalysis.Wiley,NewYork,NY,1980. [29] S.Roman.TheUmbralCalculus.Dover,Mineola,NY,1984. [30] A.Rucinski.RandomGraphs,chapter2,ProvingNormalityinCombinatorics,pages215{231.WileyInterscience,Cambridge,UK,1992. [31] R.SimionandF.W.Schmidt.Restrictedpermutations.EuropeanJournalofCombinatorics,6:383{406,1985. [32] N.J.A.Sloane.Theon-lineencyclopediaofintegersequences. [33] R.Stanley.EnumerativeCombinatorics,Volume1.CambridgeUniversityPress,Cambridge,UK,1997. [34] R.Stanley.EnumerativeCombinatorics,Volume2.CambridgeUniversityPress,Cambridge,UK,1999.

PAGE 90

[35] J.Taylor.AnIntroductiontoMeasureandProbability.Springer,NewYork,NY,1997. [36] V.Vatter.Permutationsavoidingtwopatternsoflengththree.ElectronicJournalofCombinatorics,9(2):R6,2003. [37] V.Vatter.Smallpermutationclasses. [38] D.Warren.OptimizingthePackingBehaviorofLayeredPermutationPatterns.PhDthesis,UniversityofFlorida,2005. [39] J.West.Generatingtreesandforbiddensubsequences.DiscreteMathematics,157(1-3):363{374,1996. [40] H.Wilf.Generatingfunctionology.AKPeters,Wellesley,MA,2006. [41] D.Zeilberger.AlovingrenditionoftheMarcus-TardosamazingproofoftheFuredi-Hajnalconjecture.

PAGE 91

xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID E20110331_AAAAES INGEST_TIME 2011-04-01T01:47:26Z PACKAGE UFE0022066_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES
FILE SIZE 626471 DFID F20110331_AADEZM ORIGIN DEPOSITOR PATH coleman_m_Page_13.jp2 GLOBAL false PRESERVATION BIT MESSAGE_DIGEST ALGORITHM MD5
23dd4d4eea08c9a92a4e912c6ddcdaac
SHA-1
df602f6a466755d8cc969ba3e2f1532ba240fb2c
c27edc0c8ae253644c4cf17c1484c038
ccd3b1b5cc3a97893bdeac81e13e8ff1e21e9822
24d62cc5545fabbb772630f77f392d5f
8b8b2a7b46bc9780a490a2038c32216965810119
90d928d0cfd1def2776452d5122f4b90
a5b3f258df65d3057a857681c0739a12013f2cf2
5f054874ddd1d47f0ab39ed5636c34e4
06b889b3a340b412787de8d0650f952e
7ac7e5ce04c4644a921d3a1fdebabac6878943cd
04db202fab812bc57e3fd8efcfc4c96c
f1462ececfb7a03f0a0bc6e067e578c5bc721178
0ae0f88576416ebfb624824a688aa728
4087fff87b4bf11e3662087974181ba22349306f
17f9f86c1ed064ac106e13961bb51654
e352965011a926b05bc5f47ba2fd447542afa43b
8f21a793c770582bcf173fcc71431e5a
7c0e897984449187bc821e134c1506d95e4ef94b
383817aa7965cdd3ce4e6ac0aab15fda
fabfb3e75f3853ffbdd301bf18eb9d104358ce62
db23b316bd55794d9a41c57520bc72e2
c16d28576cd1c724691f379ba29468136473ff52
cc520f002928e42157df5a0fc476b002
8f5eedf61b2ce7824d0d08ab7ecff4758292c01f
3e7fa9fe8d0c72bb4cd8f53533574732
ee5034734a9ec9a09372ace870aee1c9bc34309a
86bf991f8bc7c27c786010d70c15103d
9dbc291d256436bb413628870c59cdeb617d280c
004cf1dba4c38b6866921c85a87ae957
3e8fe0f2387e756056e84da6613b8dae8ab5af8e
5bd76c525811d5ecb24c8499d371da25
b7087317f87c5d3d35e36030f50f59de35e6a9e5
5490c06e9df0591dfc9f938ed96940dc
a231e2bf131b6585264327e225c20ebe6a75429c
03985a5b8ab0fb48db0bde49f3008657
2f910744d4a7d0e1b7743303ba79835ce38e7a13
25f63f4b3a3eaa77d092b1bf55370b7d25296719
a7191ed95249f51318bff085de24f5a2
53bdaa8640bb48009e06aa6704baa88d34c2b9ca
d34e15481f971c83b80ca6c9c4960120354d555d
b20ecab32e7cc94fb8cc502dafab0956
e4a8fd2091b1de993255fc8343387ac68c29d7dc
10a5b71409a4f0c740d763be11b4ecd43fcdc782
6c78e2ffde9ec19157bfb0a7bcdb4566
c70b63ea8f9e7792562b593febc0798c19c4f154
46da07292091c3df3a57087bedf1c66e
7a574b85b779810518aa68bc5bea552a7235745b
cbec4ba2c7971f2b22a710a8b8c8d1455077b49a
1f1bc6993abeca11984aba099c90305c
1f2419df1cf3c48d52cc23e4b9c882ba
e63c66a1aaf0ef720012d8158c71546993baac59
e409e34f12e67db5f3bab955fce97963
4cc503dbed76d7c3f6cf44e99bcd1aaa11b9e571
7cebae82e576df9fd15a3645fcb8df70
a4446250957642ed2a165577676ff717cc60321e
df8a47f15ab96f00c6cb09899a25bc95
f2b53473830087214fda77c0fe6edd116b4f4cc2
0dd2d9fd907fc712e33d973dc44baa7a
681d1e48e3e7e8126c0ce46c8d51ca2e8b3c3869
4025bb6c83174960e8e1808eed83b07a
592f27ba099899242ee344cc9655f6552ee393b2
38d7477e52946025bcf58564b1836d6a
6d30b0cb6bb4bf8df47e06e806437d9eb03f4764
da5e6d22ea159d101280e8368ab7636d
d831ea8b5a68ca1660a4c1905c2fb754f29f16ec
af090b2d702932fe148761a4151bf1a3
ef15dfd95d15070a1c73274db5d4caaa2f31c267
00e3f99d44b3c2260f3574ff24e8e348
15e5b81488c9bc487fd26847f5c18a2d84c029d4
250cb7a2f516a0991b162524cabe04d7
c94675efd558e8225973732380592c1552fa114a
d23bb9edc2d8043daebd1dc9dd9770ff
4fbd15c07e7ccf0ee3c027f900a2264f2b28c0c0
64298f6692f4dc66b27272898ed3e9d4
800606afe86e09a1e32afec3588b85178be9e751
0d4946f51bea6d87b695a44f4894e022
eda3c07c672367cfbc92791d8a2d0f8b59389821
a67710e2a63968e6e496257833b9d750
f55745199883d3abfafca14359c72e325b5bdc35
8a954fc40d0d0d170f641f3d6b202a6b
5efd596e6aeaeb2a413a06c4c56658d622d94513
faea4a70ee9a4dca4c5186398feb3129
9f34505b67bb95acaa767fcb87e68773
1639e3428eeef61a8d6d577c4193ff10f50a8874
03c373044f201b4baee5d85c5d20600c
def192f3041d7dd43ed5493ec76ca02769ef2ccd
c7964d25454201c4f7e9acc3acf073c8
fe307c7ae35ca37b5a56cd4982c2229561756937
5966db482f3dd3405e87669ea4c09568
d1eebc0223a8e5394947094890e6d53ca51dfc22
cc9479803a2a785d9ea53b0b2053f9e0
01d7e85f550459d4d22cedded5d8c5dbbc2594a8
a90a30226622475b93ee272313fb8284
b6551e229bc2bfed34c6cf4467467f53f22fbc76
fa174758c0012b8a01ae09fdefba55d8
1d6effd49595df9608f29b08dd8969e59562cb51
1d8072cea10c295770d52282622745b1
c129a8dc10b501c0b63a418421ddc8de
26c442aa2beafe2052396600d7a4577a
a89efd113df2ed26380eac064b7ab2008857bdf9
c8685ce1b116058e0a6c9e6303c0fbb0
db16fb38b657c2176e7ece21080cee457fb8cd7f
fc23317730d51fc8bc3239c78d039e1c
a2e391034189aeec7bc09f89f2958bc1862a9877
89e7066fb7882aba05ca9419deb410ff
2031d5359d0cbc9bf98e2f68c57c2e1c5392ca00
23a35c56c60dac04d3f769cd26171fb6
0627e878ebfa6d0cd3c1b77af1320ee6
5778fec7294869fe0737da88c262c5380570f06b
902129f486f7574c814da73143d0b68a
5d1f17a6e592a55254503ea48598e19500ec9be4
14d4740b695bfb166b14a633f645a0de
3de5b3cacc5bd68dc10c3e9a651073ec7c9e72ac
840a13ed9c79fed6dfa1b89feaaaf88a
b22daee0a17f484da6158c5ef46db70b
820b3d7d85b43773da7abe2a07c88c9cc393c018
cd52210c05b9f261c4c6d83b4d08cf7d
6cf21ea748c4c7e4cb614837acd844d5d12244ff
2d8c45b744014bc2b71ee4657210fa91
7cf3b292b00b0524a28e0c5d4dd94dfb
f2ca61b4567993aa6ab4fbc45d76b7af
5352ceb073afc2c39e29457ecb2143972ae29229
861a16e68d53e0310ca9ed2e0ba5d362
f83719424a325722ee8fe536414615989ff4f977
69fde34df35d9cbc985188bef960709a
aa189eb78ce41b6580cd4cff5f64fde543248755
2ca0e3c90fc403f708844877ee001a0f
7ea60b86be17dac6718942f9eb1578fb8c520d16
d15233151ff0c2b7660a3927114fd172
028aef8f52a27e70d93279d510958b6c
26f8c0925a932b47da544bdf1f310b2f776c4192
b51516d49f7dbc09273a7fd30081601e
7a44150086ab7c5fe35d4d06ef0b07d154169272
3eee44fa2a53b109ef0e5f9b14c4dfbf
824e69f3326c92f6e9a68184fa3ac6e552ee453e
bd624942e2cb9fc14cb424dd6f44ff4b
4fcf735bbfe3c08ed2632af60a751ba5a5565dde
261fdd5accfde70c236bf47e927f79e800c3e064
e920a38f1e574e5e045483791e4a959a
52dfc67fb7c65c74a3c62b6bec8a6bd1
dc45a98c08e74e04d63719485902ab6b2f057fae
a713e796f9320fc28f777d88ff211527
8496e77959f62d0c0dbe2e5aba95e05765218b08
389ddd27e2526acd1c2535099c457aa1
6a5a9cf21b0e0e3a69f4fec3df94d061
f1f08631d6faf8850e1de6dab1393486f6c46eb2
b09eafefd02aafb780da046dab829cfc
a81c45e0595f2507d2c64fcf8956c8aab8b85e07
f5fe1945db14de63c3fd0293210ba3a6
38e69d26db94ffa7c3d3ed7659d9edce77669579
622fbb27e7568fd09e4aee229d883df1
e3e7cdf0a78784211bda3f2b4fa578500c782fdd
fefa724ac226d42ce09466c46a0b3293
5592ef6e0f28947b1d2ee2291988d70c989cbf04
00bb35ac3ef0fce617c50981f4ebb43a
e16506fcc10e4c001699bd08d61ae90eaa0fe0aa
1a6cc707f148f5a6ee43454096f43747aef2b4a4
04193abd93de30eb34434f7418ba592e7d823089
db57517b37965c7c558567fd92c2fe5c
5963b438a3fd20b84b1fc0e6c431c2f6f5cdca35
c81bdd399c0618176e96c00cd178064a
16f3f296915268c403141accd15214cb5130e7d1
3b8164c2e7da1829018c57f933cd29b8c2d41799
7f731a334f8d57311217da1d400e4aa4e1db24a3
53f05edbd6f75694e6c7802ab374533d
eed8cc0cf1ab7ce8abf387650e8224b89cf1f7e9
5ed651a3388de103c1246cbf92276753
dc9ff311520b629ce567b42d3b96a4e1e204c39c
afe03b3e49b92e106df2e9e326519f37
d5ed564f55c4b66442f9fe7cac739e907e5b9164
5cc0b85ed2299ebe3cfb7e9ff009bffd
86fac0447e5d713a6d906dd02ee571b04910066d
f894489e8ffdfc135718cdb2136e64d6
10d54e83190d3c102ee28d1d92cef41dd6c66cfb
250b585a9dafa93ea19627f5bfafbdcd
fdd1287c39eb018e5ea3e18a36a174a9606f27e1
a01204597bdfda01482e4e1f64cefd4a
d6f7e8f0b9cfb8323dfa7e9b7084db5622e616be
ec2e053df3851a96a7180b413f028ca9
4019ce3f95d00007e6789641edaffb078461a378
c3eb2c4d3ec444c2fc59e8cf2f47bb4d
10c2943125b80e1f8a8006276eab37f9f5fc350b
a85d2d636bb5c0866538bc5cff7cab5a
f9fc28f1d2fddaaa296960c2a51638e3
27099ec2c650e53d259b95892d53d8ef04e64122
3d993eda6dc20f77f7bb652eb935cb74
52dd5d41525a673477a42b206b0cb82098daee59
558592cc9cc9025fd6921fc94724d21f
8074cd363db7066d6d0771e1c36ebe918e094a1f
305b925cb1de36c6f2283f210980906c
d4e453a2704cfccf96d5562daf84d780
fffe4b6dac6e2941bb1874bc56461f6537d02753
9f2ded81546574861c27487022cbb8c4
02c3851a0e1ab28037f9f2ef93ef611aa7715b64
33671103aa9ce74b920367e0e9066bf2
5d54f9ce296d7f1bac377f3ff35c8ef29bf311b8
d8aaa3c7df022a17a91e61320ff9e3c7
1c0fff4e0bf73cb153cda1878f31975527308ed2
2fa1ac28e46e1bc4996143d844bb4bf3
32c06dc61512b5e639e6cf620ee74b79e17b3bfe
498126421fa2246b1327008b5d580783cdca6be3
cef8e7d6f43bdac469592dfa77aea227
66d8a59d208944461048672741529823fe628e83
f4308f15ee86bd3bdb58b734722f78b0
0ec10b8e6800e7f719253a411a7a1f9644d32482
71c51c68baf20c0eefdbf3f2546f05cb
bf87afe3e294b950f28a1d3d1de7533b
559486879aa5def2a27b46018523110c34dedd8d
b424cccc208084885631021467929ebc
3b9344be8a10e833b0d58692829ee3dc69250ed3
1f0e7c640d301ced3731c16009c6d73c
140d31f219c39193536466c19a6482a77abd9d8d
b23b5bf84941e7511dc7c1406fa2a80d
ca020831e22501a2c926dbe07ca16d3eebd6f9d1
1b51af917a63aa479753f1d796bc8210
6654caea700a6e13ec1896c72f3842b2
0261fcac7c32997b06b8195e970e295e1ea0d493
fe322ac29c051a1a0790a59c87538d9a
fa89a79366643cb79f7be708f9c0a72066fa89c3
c85b671a865bcd8f35cb3e93bbaa7e1f
32311a2e0428d19b32f09b9b6828b4bfcc472cf9
8c1d8ffd6f419ef78fd9b330bd5ae596
6a061d297328e745ea621eecd6f76d23ed7c25c5
a1a601507dd9752f6c093894fa4f5070
c7758e6f9c60a48209abb59d0ae55fba8d0bec11
c2c578a3d2b6f139a7d2a29c093b7764
2f5d0caf7a011cf71d6c4c07d4f3f688e1499b27
7842bbde51651141bb0b441afa9134d2
89e816556cc0f7448917924a3733f6fe3a9caf8e
5fe4351a0fb5b83c91937c3634344b74
ddf38663120aff785a2bab2e06ce5f8d
f9a5f7a4cb399a5a6591f5705cdf4e65619c0521
0e8cbe7710e09dfbc8918bac3421189e
808806ef75c412770afce07ce639f9a94e885cd1
a5cfa086a4f2c99365abd530dceb094f
9e5e83f15d1d7b9bfcfc48baccb85992504c655c
ede2393c511beaa9a9a02c72aa1e990a
0f033e3fc82981a91418062c7ecd5c91
f37b312820d0107da49d362ca49769293e223c49
1a6230eecc11517e86dac82c8c4f5e9a
4c5d7f113e898b42638e7cded70de116
18d4ac0731abd69bd2e5df5d6963a6fddda72e27
7cfa917ca1b3b0b82347305aa09e83a5
6da840b755f168f84d158212a2f924c442112e19
10cf953cf8c5e291e3dfcaa100e60884
4f01c3a44f9877b5be4c8055c990ae73dbe3bd83
8709a77ae7565fc5b3a0d569984725fcedae0b1c
6a5c92e676838594ba70e1b19ca4ac11
f74c8b4c76b2d5b397de23f50ccb6884fb9063a5
c993c2ec10ec942c933604f77a939c78
056176abf08b8bd3abaa193e1bc41d0435358ce0
5a38bb90ed3d3a70e1f0f84fa1f6385d
493bcea6f41a61b19b6ee2af4186b172
920842d2f663ae95e5265ac2e8328b19d29636a3
1b7378e5a8ed08f951e1e5c1b1d4250b85560d1d
1ba37e1a6e4e1310d8633eaeec0a20ee
514cca68011b668d6df0c4ae8df43513247d5b12
3e1336638e978292f2ff68dfabca622d
bb7c04004d0eea3c3f7f9592e6f32be1d5919958
8dd659e68001a98b7b054e563f9abebe
b6552f9840b9550416a137fc6a7b9b67429bb178
8bddb4cdb163844b21712d30cefa0976
2bd0b55402cd3abbbed72793e7cb8cc89af70f82
ea721093a0f791993e0cf4e73eb44df6
7979eff6b9c4acecf66da18b0c723308
2c6fa264dcd41271fd4b001c278933327423eac0
4f9d2bf465377b38f43f6b94ff363eea
aa81e252c4e5284a747c759b8219a9809fb05b73
59f7ced9cfdd3b6b49f9b048babe9c81
d718418d9a115942a8a551cd8313e68b7856843b
3ddc9d4de7d52c607e5969f7b16fd1bb
8942a09e7e6ab1e40f92a68db77c7ce1b1dcb6fd
465109704dc16190725fa9dec0041d2c
2e0806586e8ceb1695f12243ea4558d1a24ac0f6
f3341f74d2d9a682d7675836c18d827d
01f432c2b21e370eec395f7764a59950421561b0
4c70fc7ce2fe9822c372b976c23a5331
eaa0e969268d2bd8772073d37b2d3f1c84f7635c
f4b3704d094fe885aa9cea14282926da4f941a5d
7e533733b883bab04d890269dc0376ba
d6fe1d9562409208746e8741eac6325972b493da
0f941e70615cbc0ab8c43d6d04b3c26c
5fe0a230c7c441f3c942f7c58520faed25948080
766de656ec2f987a50ab55e23cb2422d
e3bb4248dd49aa5cee55790c09b6638e5e99c3e2
6f4e844957637efaac38016c39c926d1
80c4c2056d7d171cdf90aca527808f876dd2c4ef
5d0b365809a1f19da4e7fc9db13e06d5
fd8d2775b4969500291bbc2003114ff0282acf77
345e807484220eac9f388081f4b03ac9
507450047568295d32ee5fc72ac5bb577c1c3466
65639d06f4d2ba262538ce047cd465a3
8c3857356e140fa6879e1863e77264994e10f749
c56507de702a797c3f31206ab013aa16
3ba82fbc21dd2d97f28830730512a2e4eb54cc15
e2cc2531149aebba9c0b39e0a6194f9b
562b47ba0041fa13c5e6fc55937f1d7ef612aa8a
afc7dfa9f43763b7e5d6e760ce58589b
b806d5e50e0c4d932d70cd1e35d93e01
a93d3dfc9ba2437dfd85d0b2308a27a4a5bfeaf8
81e06a36f9432961800b60e140a5d69f
2797cbdf271b239678fc4e45fb5d47794fd2d090
4b7bdfb8ececa6d2e7e5b2dccafc51b4
90979a2f530a7c0cc003794985c7d888961c7e66
29c9194c73e41a9675e233e5c87afa27
91d9e3fd9ae6e5316b7b6574d2bac7ee436ee7a7
24c6abf10d6169e93dfe2131bbb471fccdb7cef1
fb2143f6acf6dc2d0bce8661588ebdaf
87c2f4cc51777c0cc115719555e1385d70c2ce04
dc25817de1b4e372562b2d83349d6f6d31d5ab60
cacf9743261b2e5a0438fe0ae9f9b1a0
ef1bd066eb06d575e8db2eeaa2a0c0c6efb4dd73
28b0cdfa269e7969d616a6b112a40901e25fd31c
bf8c5ca196b154aa374609c846bd991f
f737cdb650890bacc4636e85b123d57f173363c9
26f441a23e8efe22a0cc3ceac0c6139e
24b09affeb582f69886ae341dcf17aff12001bb9
94beb74e805193fa3ea2e8defff6fb7e
82e9cea9fc92e6d4d48614d49fde39abe4ebfb1f
cae2ccdd113e6cba0e358cc460a11789
e9a1580d0bdb7057184a481d6dde94197d137836
6339db2dac98b495dff9ba1872cca71d
cbf79ca0770674d3f540204ca8e9916312d1046b
90987f8e329ba25205d6e472a7523cdf
f389a4870d55cf7538c827391f0c0f364dc6fcce
f65675d56887d7c9409084083262da10
c0012441cbe0e48ee69f6f756cfe9bea47b11ede
eb505291be5600baa1c3f0024e44687d
447831f0edc7ae501f4655678e676b22e637bc0f
dfa773b9a5bd011e2fa33068d5a7c229
c6a472793706bd89d08dd69e7f900a29748ff8b5
1cf8e60fc1c4ed8daafeb6f280b7ae7c
d8d6ede7419c8363cd3c6cd366a633d28562b86c
142ac47abcefcc3c600b2600b3ee2d75
2e9cac6bfccf28e0483f25e36e4e3e337fa14473
ac9157aed56f5858656c0899ae7a612d
49494c02b9440cd39bbbc0d5e26b2a3234aca840
ba9192cafc7fb1674b2ff83afb55ee7f
b5fdcd0d454d402ba2219929fcc9c94f
bc9c6c5774dd963091cf61a0fe26ef25d58654d0
fcb56b158fcbc885d33840d293f70d4a
b1b8f20c28d5f6d583d1682e46ec81aed53fae59
7f202ab4e3bae84af90e54838b79c986
7e4e61c1faba9bb477616949e9e27c85cf849132
0f38db45e76f43bb2c4bb4413912e92e
389613a6a392e1d95c405aa2a28dae20db1bc519
5b83d6752f773d5e462671078eb143c4
e59823033f4117816e6692990b98cff60b1cbcf6
0253f9f650d4e80fdc97fe601623b836
da64f0521e8dd7fe2643797b23e8ccdfaa1ee33f
3521aa4c9fa1a09fb83f820ec0bce238
5082b2520c51faa1fddf25ecd2c8e280bb75710f
c91e8d71840f5aff3646b9a8c0d7d67f
84764ded7155abe917629f3db008100eb2e68255
1e63c2de59478ca1d1e73ac30fc8e68e
023dedbf09f59d0e7faa169385e3abec736552db
1da461151d2eca8c992f536d88bd6201
eaa63686e015ab5e8046e17a1c51fdee77608d59
1096c84aa68370204803f1947e2ab66f
9838b333691755dd09f262db0a78e6a0b4387e49
421b59056c8c71eff1cd1b81ab897b41
aa059cb705e3fb73f971006bf20ec5422c0afa59
1ea0a1b3f41f4fd1a63ebfeb6afd01df
958b13b9c56be7b19399384641c4240a3885d647
03365b1bbc6fb0c95d02925c2e19395c
81fd6305ab11b044fc20b2247a0775d73a9f869c
ba2136cb22feda1f7dfd4654917df58e
b2ee9d40cd6577a515f1fb6d4f52f890c809b899
e93482e07b0b3b396353d49eabbb03438c475b12
b9f464bd1f47e206b5eed60870e47079
8a935a662f6b254949d0d3b17cbbe390960ed859
a7a4a731b1e69e64aa2c51de6dfe2ef2
37d693721631dd3d8a61a68c83d628f584f485f5
8c92e1b84b447ab770d9df6115451779
46241d36d3d6b34f0108c7e4e48896f15ab66ff3
147704 F20110331_AADEMT UFE0022066_00001.xml FULL
7f4e120b8e7b02e392c65d040cc0b15d
9ca664a57dcbf301827f4839f11f214c48e75158
f87f218d2bb3f41bed2bd02480bfdafc
c62b0f7b0d2e7256a61ed97399e28be0
5c4067beb15f31d11b87338719f4c2a0b88837b6
8b4cc016864e8bdf9f56580f6ca4e5e3
95a42b58e60147ca2e2294ee244c7a3493fd8a32
ec9688896d24025f1e0a86a22052cc3c
004b9d48f05c205153821695cfa4b7db09c67f32
79f2d6c0b053c3a8daf1e254acec428f
605b5afe71203ca08097b4dabdac132bfec1db31
edb2922223701e6646b7a66d591a549c
b553617b4f400eca122c883d3ec5b92e9a71bc0f
29a3c21f609e4744274d6faf8cd7c6e6
2e621675af303a67a8690b8a382290904035bfff
0d794efb018d60e2a098d65333dbea88
eb01e35588c674ae6b224bdd8cf50254575b4f63
36c51a88d290993142415766406a0357
e3e4b4a0d9354da7d1042208b27df009
e9999cff71327b64d09e0d81b22ed88d8692476f
77f4a45285623c041529b35c70d56cf6
0c8a0082e02f1fa47b9210afd717ff32031fd9f4
cf5ae8dea26cf5d45672a47cf71a0cff
b7e1c90d2562f567ba359817ab5acae164058ac9
6fed8e385bdf7f3be6db0931d4cf920e
98dda9cd02db63d6725e5c0002ba74feb34507ce
d882cdf004e560ae7db5932acd6910e6
dc552961f11f2c236b4a345334d6cd41
074e3514040c473b77bd4b9d79f75759c7f5858d
a3b1541c9d61cbf4c5a3fc266085f1fb
67aa8e41c42148e36c95088ce047b805cca25b80
1e8d2a3bec831073e2ae89cebd31df9b
58f5fcd655d4d8a84ee804ca52fbd3ea9ae19e7f
07c2310eabd9ac46dea76de071876f68
f288f961114f4427e89de521dbf5d08f819cbfce
e39be2405a547fe4a879aea9f783423d
bb144c9a0d8534894fc95726664d6c2741de06e0
169ff822ff9cb431133b0bcf1b46608e
8ea0286988d728721e22d7886033ab0c
49d4e5ecb5d5f6a4253704660a3d8b00b1890c7f
a579018eb98915bf618125f9956aaea1
6172528148b8bdeb57b536118228ebabf3982a66
5ddfed15fc608acce6660de3b3541788
657253de71ef21f370d215c389c3112da5d4f388
25c2f4a357b08c2e6e22276a69d9712e
1a7c20aaebcd5aa7805cb88d6e68d364b5e2f0a0
d45367625185f37858d721af0d36e007
e7a1e0ef330dcf2b114d978921f67b0c676dc4bf
fc1fc45b734c913df770e41f22c6bb4f
be81cd4e5ba8c010b48a5c976fc539d12602f345
283e8b36cd4a9bf5c6e902d54d33b9c3
e93a84cd2d2ce97272295770939547c031c49d8b
b8339ec5b79eb801d0b454da9b0f90823aef7bcd
25a79e0d7d8c375bdfa418541f2059b9
271797502907aa1b52401635a1f4dd58
d341bf51f140cc1438aa76a9f830bd1f
f7b6d97e2131746e77f6bfdbd9c8b0a164a13e74
37e1c8ac521c8132110ec0ae249af6b2e879d4bf
4ddebe92fe10f4fd0c486ddd2af79e86
d4568aefc266af8fe01af053cb22372f5cb406d6
35f1b1b322880766944553c07db0944b
a282437e67c5ef91db0c7b839d5cba21
8008219666d50abacf5d301ef63a243c804230dc
88b6bf8beca5d43678242a0404e0cdf9
cf5fc12c00a10712a97fd54f87a8c30a59510fcd
8e382e569315b2f17bab81ab640c57d5
975b0c5e6d24f2e765b8184defb7ac4c6e94c10b
9e6d7403aa0c0686ab06e77606794878
f672236b48dece30e85c3f5b0154ddb0e0b904b8
cd4b5e4ee24c7b3a2ece684338f447e952a75bcf
e729fb6570cf5e31982687c962c0b386
ea5b8f34653a46f2425e33212df404d06b6234bc
90a31f175767ab89b50ebbcf77f7deae
bdbea2098c93e281d8e3d814ab3ef9aa129c8ae5
87f0e15aafff06474868c88497def41d
609ab2b5f23391430de6dd5b896bda32db1c4c9c
f1ae46f01c0e197f502171e5fc4d4385
a80662bf0326c6e6211aec660a8890e1c34b7c4a
48b4de2f21c94fa52e5149e7fc1f0ba6
15dbbd07ac59a1d932203207bd0e6723
d50b8a35f91c8aac441f0affd433c4cc2264b017
a80a239ffc4e465988d0cde95d780d87
f90bc9c5ed1f3e081dd238ee3f1945cf74504fbc
408f525d4041c6d063f67601fb1a0514
5c11094cae013b28541e5aab9f81078c78a6405a
1f2c3657cf4521ec42642601feba8407
7a1758ec73db6f97984f90148a9ac71d
ff7b3351a6628bf4989e44ac0bc41bae
26249456b2326917bc73978407ce3e0d80e971a3
c5eb82f40755795d632b445e1974ab6e
6a13a9638c076c6beb80f2b7c61f2e5fa92d5507
1b41eab48f9be7857d0c00c838356871df47d5ac
229dc3ffa3e5a21e3a9791b8942ed750
25f6b0f177336a48e6b90be318b7f524c8e51e65
7ab7fb1a2ae52a96127725833f01b826
6d4f92e07d83b84e23e0deb29774044dab54e48f
8db3d1ce0f7369102f5087fcf3077465
16c76e6ec7e87d14b8c0ffd985d00818
55f2090fb55b77691c5850e73f3b60af1224bb9b
cdc9138b3dd1c606d046cfaf602cf354
d029900a4dcdd07a1f0d1b8be06d32271a0647c0
eec5c9c49527080ea7da0715cfaf531e
d4de3de94812595fda4e076dd71c491ef00cdd7f
ef22e59a9247da8bafc59ce0274be3f1
77f0a4ff0fd2d386d0434e11b6a36d72912f8aeb
20d5e01a1e160ebf437b4b230ab1b0d3
bffa5e12c0d43260a6a544c3b00405162ed7162e
d9362ab8c4fc0f85c018c2927cd2dbc5
7f0d4c995e59bac8ffc32bd53256825d042b4d24
3b8ccd757a7e292c0cda5c8147f4f417c395f20e
6b1b2cdfd19b13b70bec73540638f80a
b815714eb5ac8b621af489c98e9de63b3dd62db0
eeb89675294005628bfa74725514abfd
88a4a941f1767ff89eb329bd3ec99ba4cc4a2c8a
5c56e72a3764665cb1e134b3b0c7a1f7
cca7a3674241658fc74326dd4ca070b2
8e26d974cf91d57565bbec1bed70ea9f3f0b5734
6ced7c127a3471e00e5646d124178b0d
9968f5a87bbaaa63589907b835d76b2661e9c10b
6c9c3b3005f0c4ca2fd5da4cabd91b5a303f7c65
ec5981c87e096c3b530e1201df47ccfc
209bdb35e6992ea3edd2a521f6f45d5e55964322
4623f0795a9b5de84f6157a3b58f4059
4c207262cc668338d3a8d38429e41b65831babeb
f2114af749d3d58a45d4090174940c78
a14740f5903d0bb93c1a8a5fe9d20a8ac02ec3fb
67892bac6d28d5c2ed81ac9ba76c7d0e
e95d4113287a9d08721e91f38e8663367abefded
74c38b6d48959c512b7327edd1357afd
269337f53f8199baf8bc812544ebe7014f33428d
91b573465f18bb8bdedfe017faa61ddc
6c32cc4cebc595fa7a4c108a0cba4ff759ef3fa4
17d8f7b6e84c6d7e8ab758fd7f3a4b69
95ae9bfe6a59f96d3a24ba6697a144bf
f664896e397a8e0711394c083bac769f6cd0f353
67ea69e09158bde6e11cd2dd2384b299
f45c079b60b5faa2b96fed6f7a07b98071dd3e9c
242ba9c3e17351ebbbb6c70b6ab03f20d6ab23c8
d5e6d2dfed9b4f832a5ec6900e81c503
5a39c323dc561bc12777f5f3203dfe84c4e29fff
c2284e51679eb5699ab553c03ecbd61e
4392405c95e32bbf9e809bd83dc2382d71539ced
e664d2af2eb1556e52377f82754ef8b753f372f5
8be207bc2027fc40198c9bd66cddc224
0542ac65fac9e487dc7df5ba3631439ced64284b
a0976f5dc6fb68a4c57d6cf95b187dcf
99877c6fe5dccefeda3f467ef59d0bdef9b16a86
d801515013b1709ce577f741b63e831d3b6167d6
50b1dbfa4bc801db996077ac94fa933f
5d5a69ae9ff40cc1ae51451fae19ab95a16bcd10
2d582b2a991b3ea834c8967a47aa1e11
97992490b090e94a854287f2ed05670188601b83
24b9b08a8596df05fec24be74b69633b
2796d324aa770da7054414d05b005e1ec302494a
2eefcce3bf8d4ca278d1be8db72e5902
d19cfd63bf929f7c26443efc4f503e667bfca5cb
7d9f6f8eacac20c57d3aee1cb37a6c5a
c866910d1f03f821e16e37da7475bd7132d696ee
f4183f0d1d834af6e4b7c64b3d3e7314
ca18f513816800822fd5eac23fe254c2
3ef9642e16f5f3f9db897d3d184e37a9e596bfdf
bd61b43c5a67a877b0a349a7a5308995
cc3d2e0d2105dbc816c56298eae39386
ca746b6294cded0584706c02a354a2927b02645a
501d5496a66b3c170fac222d051dc3b2
762068730ab33d216cef48f7cd2be5f33422b16b
5ee489455cb167e3dabc22219878583340d44c4b
bbd9cc9d3c340e7cef32067f547f83fac456a834
fee67f7e20434c65b6a50e487c4299b1
4f1efb3649778edb8201af6a9e879254b123f653
ea296df74e9f358abb6f59947b273fbb
451997c4271511623e3657e4c49401b0d0d8ce7a
e916a642fafc88440e68acba1a7e87fe
60f999f2ee063fe22dae40709ed8d4d46f8900d7
444fb2c2f72d39e752c8d3cd89a0e79ac58831af
3eda0a6737ae41b568dace81073ddb92
72171eb768907e4acbb88487ba0fbb7acff1682e
d77c05fe7012769eff3a6672e3740925463fd645
0aab8c960f89fe5ae5976750083dca4c838e8a48
b78809ac1a180cc35435612d60e719bf
2f1517c81d494f7d800fdf011f2e1125d3ee5182
4840c93b9f7ef155ed1b1e235ee83ef9
efd419a9914bee49beb2c74e029c7fe044b6e7cb
eab0af2c15c59df69f4778b0dccb565e
a8cb494fd20a7a6856cda57a6607d3c532542540
5d1e8c15382bd2265abcedd0c7f34e93
03005f9f26248c0cce8277c1e2349833dbc718b2
9fd595b5469a18a0b614168f82e5ce0a
183d6b386e74d7169210b67b6f0e05308479c36b
49d01e772dc1b2c8d18092f8ed0ebbd0
43778066a334ccdbd7504d2d8590c9e5
8a5d49e821e35b3e5265308524411d2b410a4d9f
f5353e8562f3ba9eb05d38dab806cffd
2781fa895ff70db493a26e265543702ae46af4fc
8b72071dca7a21e7f409322de9784ab7
373a797879392b4fe4d719db033c61f7da0952f8
cb0719dbdaec587fc5f5a939cbee045e
2c2aa682c88b6fac64b1e8368ab8ec0b9817bd72
b49e3436cd7554ce369986826b41b810
8aa6132fd7c3587657a0a54d48c7a94da16620ef
1d4707539c7f559994dee4035aa6a1da
6fe6cb694f60bfc36b1605657b3d0fefb51d6d99
e92b58d60a0b33698d219b405bf323c2
648c706d105c35a9b6a58e90c30c32f55585d830
226ef9845442d7973c3b52c5084bd836
806b3edd319c2ab2fb506169cb5c6412e467bd3d
ec6fd3acfe71c1171e2a7a731b55e7fe
9ac69291e8bbdd618ebc2e67996cf4dec2aaefdd
88e4ae21ed12565b077276eec9f675b240d44a34
6dc55a7970ae47ed4bf7729a8f8e25fe
e7731e799720122d9a4d01ac7ca7ced08741d0ae
0ef016e515dc2d22807b41bb1c02f8265025b5a1
21683fcf68719b981c21879072bdfa4c
6e01612df135112363059cbfc9628f7fa927cb74
f1514acf3f5b477a2a1d1f97e716766b
81a7c3cb111d6815dabde3e1e6c72c821ccf7b84
a824f7da3fb92f0f7292f63f7444bca0
44335f0aa915852ff97053d3376472fc62fdd0ef
1ae0c07ab55f8043041d5552553fd865
e089f01a6c892316774eb05afc8e29908c8bbc8c
dd48647f6f72f4ae178ac43e65098b43
d209567d82a718bbf198c72d54f804d9e6d40390
7852b1d8e305be29e70007bbcdce4e01
9160d9733cc23c2eef49c9838e4c46b7566cbbf4
762a67f101141c7bca564e4713d7099e
becf0087b556e7fa550f9a91e45480e487dd93be
d90b778b2138cb8d9523c3dc987d43c4
68e310350aa1efef0383170da86b243e181fa896
0a4afba84f3e7956cb658eaa1aa20683
20ff62d6be3c055037f7336dd69bc7f4840d7abd
10232405c1b2e8b6d06e1d2832b24192
e37bf20492202454ed0a72cdc21d3cca03bf3990
6b0a33b77f71d04bc14c9604e89af887
808193e046bb12b2ebd2af8812113c51
c34c7b5731694f7ded0a7e2e6d151febd592af24
1a2ede87bf8ef3b3cb735396089e6a0c
1f238e32871ea53aab6ea2b55c8b3a066704ccc7
d6daa527d953e0eb1b253f952d8d3950
d96167428b7500348ffe0cf641bf02ec
6140d7a44e2e031f7cb1d7c6bed9a8c37d18a526
27b134dd54cd2b80acb56cf98737f640
830353be1bf022204d665a87cf4390c1
66614582f25f3b6700562bed739b3a67b7429fe1
8a82475129732d160b4cd4db6838fdc4
f1cdb3fc2bf12c3ae5e6f2b2365236009767edf9
d42794a9584bd6827fcebb12193be8fe
6381b0380c25eafe5dc0c96fb1bf559a0343893e
216af00c3ebcbac436ea44263b01c72ee2f45081
8454309cb8b6c08b7ae49b002b606ed4
88328c12c0bb9eb7d86bef6bfde2029da72e5a23
ec86ff23b9b1b0afd3613b5192d184c1
54feae99a41db96b057a89121ff29d1a9a3145b2
56fa6fce3a5d75d6af50b20561f7576b
173052e5596558a23786f9abc726c74a
55e46b0ec0010dae8c4036c4fa49704da0393100
553bcdb1fda96e7bd286c7b194779022
71e7b99ea43b9a40e610b1032e2c0fe3d57bc0cf
5180e596c5ab1bf625eaaba0a8c4b87a
5bdcc7c05f49ab37ce7625248e48aa0bff55bcf7
12f26ddff36876e687780a088cb7b49c
cde6213ce42c2d38b873ce77a18f95b1362fb94a
217f27f98fb01f2699f21039ac113cf3
1f5ace740ac5625ec27a58fcd649b8258772400a
818ce247acf30055313f8e565bd2a2d6
afb92f833cf021b125b5c5d2930c4e696d2752c4
0ed02517d6f31696603d21eb88823f8d
7353246acae99723b65cf4680263a0d79123e0de
574a48bf0d0864e199836fb9b9a504a8
1ca2de7f5e871a7e129110423fcc2ed431790251
cb793de95101e49dec510c571477a7ef
f0dfa89f3a0593ccbea7f3d5ff38d817dbfffefb
614437532e9fbc7e9d98fb52d139af606fc3be34
53600001bab3c7dbae5bc240394f87a3
084af8c5e27fe51cb8f0840865a1075369158184
c79c536ef71a9c92c8afa6c1e6104506
4fd50c863328460441c48e0c6971aea9731f1ae1
201f66e46535e93f54505a8a6a2301a4
c859d0bb589063b8bc42127db5cc3796
a11a0019d311fbf0830072fc74b23f9996a81799
8b83fc9b8d7c354f89975fec652475e0
2ab13a808cae53570e2ec779e84a005de692e5dc
5bd70973102aff1e3fd5f561b5edfab5
b582438056701b48f6b5d2672cd1a056b8e78b66
7aeb9bcccb498ed6438198676a03f1d2afe2dce0
9c2ca8d31d6954cf1d7958e557d7a27b654cb7a4
5d944525fcf7be884b5490b639ce31ed
c004c234be2618fddd91da805d8aeeb6b128371b
6ff354819abef64c55e3fe43a67c7eb5
dcd3e4f2d62ea3fda6f96f2405d22d4eac34f881
9541176c6cb468f192fee85c5e6701d0
201c532e2ab84051cc9c7cc5931f5b81203a98cd
d4629877655d04cb3649d92574f87c74
aa90b601b3b2414cee3310f0768bf87a5a7d0f08
852f5821b0af811a48eed0aa5de2886e
e596b79623f7742f4c532511638401b3d105e3b0
fc356ca8bcbb05883291c074798f6a80
60dd3e89b946d7b48ef37a4ca9a76df5dcf904b6
dfb29a332143725403311c4c9b9c2901
388f9f54de227df497d6cd4deb84db79eb67a43b
5e9f5ddbe479c9199cbb4404299e54bb
798ed36e52b116713fd51dbb2913d372aaef09a2
e0f5bea2cb3d7c958e0e89c2eae13c90
30ff75226535c58f25e5cd82e3d4198743274fb1
2f005d8af111474478479fe20101bf7a
1fc3aa29c37a94ed03f15b68eac3dce86ca5433a
f28683c0ecfd08fa1935c92aee28fc7d
3020afc532c0fe1af9336e10a42d6dc2b637fb81
430b2c24aea6e64969334c3954868146
b8a00d7047e8b35c470f2f8dea76430da8907500
b27f4920310c79f4b021627d80858ff5
76fb554615982f805de6625eb79d768b39010049
ba3a6c18069f5a801a9594d019f0a573
c20f5ec0450694d0617931bc151c16b2210b4af2
42bce7b5d30159d7ba859dbb6f0488600ed2f067
2d0c1bd6b47fa80a23b1c2b9f59d07e0
20846abdc69d4c915bf338e993f8acac4283590f
e799e1b9645fcfe685021695a4eff9f4
a5197b7a5bfaff9c662fe211641d0aa97cb7b26c
2ef434f2f0499d31cca556ceab4139fd
b7719ba7d65af9942ee42c3b8be14ffb
478eaebf026bee5c0bd731607795af7ccc0d572f
00430ca926525738ca6bdc6a3d9c1d42
fcd3e7321b0530aa25ce94310da62f1ea9bb508e
d232541e1e4b35f6e5e5595c1ac64d42f669c394
e32a991dd8ca5e11d70dc38edb1431ac
35a6a3412636ac9ee0872797fce079d648869daf
27caa5e2597cdfee4715492bc7193f59
bbc804ef56370890a88014bbf5da3449edbe95ac
9eeaea8a6eef06feaf8d0e6057ff3504f7bd49df
641a039f01faeb4cc0bce33666d21da6226e8bb0
8aa2e5a5bc07bc96a2be4334c13b693a
0e0851c03a6f782d9345857ed8df509f
224af84138ef0cab509006dd75f2bfab4a316136
1ee31a6dda81a7007e32fec27fb1d2aa
56f0b2fb575069ec95242109e383b22f09697890
5c3b3a811fd54877e56a81124e5f342f
68ee724c639a432a29ef9cf58a7c64836b58e30b
163c2248fb6c91cf33539b3b927f9d2b
bc302b5c079fceafc90bef0c48b86d030935bc7c
ea3153e5ce21907e9e005513a7b22955
08f3aaa704a2e92d431139dae915af5b76f93f36
5958636ffa71d677f23bbc6660fdf74e
3c4795a419f122f446ea2890f2d8917590e2665a
c72b577c5c2e74f9211333c022643266
d174f227793ffe9f6f6e1a50da2c5d8c160279f1
687113286a76cfbfc17dae03592c7dff
e874d7f64a65a398a3988164c77e1d29daf9b6c2
66b6ef760b56508bd1b864e231c29601
99a0ea670d85cf4608c9d8396a8c1ff4252b6ccc
d3ecfdcc4af2d51dd421bb233976141c
27b275a866533a0508a3ac3d606792e44bdd4084
244fcb6f7e95afec9bb6cbe40df16b8c51d72d41
94062960f2dd8894e23bc6befd3c46fa
1a289a9364a8808c51634c3c1fd98212
ea0c8f1bd1edf9d976f02e82a2e271c3bd355131
fbbebb5aaaa39aa369ed05dd71ca6893
aafff9ec95ee1798c3513d14f63c3d4596c123e3
9d66a1d1c4a7122571954477b8558e67
47415cdbe05c60e6ab469a272505f56537b3afbd
78318e88f7932b8ffc1dd242242f9f84
0409b53b19708513c3605e5d9902ed731aafe032
dd0e56ba145a3ba9d9f8d0c10b75575e
57c661c5ca268f0b3bf517dbd25a97539b66a3ed
1ea52ca7034f07d17271596db162a343
7c0c8faa0f59ec117c788bfa426f99cef0231b87
f57cd9098ec2b4a668311afe01baf37a
8d1dd37f3f7b409a8e35734bd23b01aa1cb1bb05
685394f40bca14ec62bae54c4090730b
7851f78f525f1b690003f080b942d20ed249ae84
ddc35c52f98ebbf6f30ec0599cf31379
c0e5269fe36e661c1f3064f9371c89e274d240f1
8e8448603da20af9fca291dd666737e1
496d7bddb976bbe0b1f4fd10af9a11da242c0a71
66aeb9b4187bc13e0d9e4847de5f21e9de330539
460be30b1a22a75ec50e2cf40b72e831507d20ba
22b41a14cdb0cffcb9275587166ee6ae
570b7a7fbdbf71639385aacae6631cc821f6b463
081ba7fabff5900216fce03752774437
5abf227898ac5e17bcf339dce92f5fbdc8c184bf
8a75c3aca642d23ce721afdbe8e2c224
22e2480f8ae60e5f6cc70381d6dd3e09832f2bdc
50caf0c1b9f3e0c769588e4b47b3473d
2c74d65fc0fd569f63e74710880d498ebbe43487
e618d1d6783d1c8db096ffbd5e5df0f1
b036e6e20ddac2dbd81a88822f27af6f0415548f
1c66bb373063fb92f1082d878fba0ed8
48d93df963f5f1e88486250ab2b316ec
6665d60de3f3b562dfc9249ff9149fca4d8b17d3
d63da849a3cb12e4aedef0acec141413
cbe034c022c9470fd7590ec9490632c3a3137c99
6336b87a62a6f6ab960014dc7d3aa474
f1f5672fba2607c476fb77434df8e76145db394e
425d1fb6ae2429d045e776de80221d41
05efbfbc8e6f45960ea7f65bd0ac76b03a772c31
ac032d8fc1369cae033992c0becd4a55
5304920f73d6d9f76dbddc5054c3e6400d4f8484
d56366ec49528a7b816568c2edec007f
4949ddeb3e29d11181c5924c4c3d3a913070691e
fa6f9f86996721f1356dcb39fe7a3532
693461f909c2cccbd61faf6ce00c34eed6ee00bc
ce43040685120f0d003462c03874fceedf4052d3
1feb011d7c9dd26a97b6e4225a2f8a15
a904312d8298befcc038a700a9909e68
7d8187a38fbd9b9ba7f8b04488ea523786104596
c05afa2a200f1a60f8e68fdabae197c3
3bb5227ae98f8fcac86ca4a5f254cb3c207677c6
1b3a09cb322a7e8108334443fb91b49e
f7d74eb6a53a08400d76bc24a125d1d1
c0662743e00d88276dd6613850406454a6501654
6589e633a40d1727bf35d3df5e439ba1
a34f7d6d6b46850982558edeba83bcf2d0e8cfff
f43d08ff16878827f129580f115b2794
d19be8e9c937b540263f5696c7698446e3b8d8a3
aa96e8e274b8bf7631ddabc5e186fd84
ca62d890876a9bddea6943bc01906a0164ed490e
8ec208ed731911f3dbf604a1db1fb47c
55df0f940a4ba0072fd8d61582be5d4dcc55af5a
105e626744bf96b185954ea2fb3b9b1e
0b776a4c9c350046fa94669c31e4f4da898b6514
e80453b20a190a4b6205d148e54f8265
63fb027600b30ea511c1e7ea0552fe3d
598170c78a124322feb36d536bb1c3fd4be08465
eb0d3e24527bf4c04cf8b8f499a4bcc0
b69ca7b69838db38aec73e455c00c7e627b3bfa3
1ec9ce62f5608c80031af922a3c82e2a
5c8603e7265624c81d327b6e349578cf6808ed16
1aa84a0e78bf9e610d4f646dfc9a5e32
a71cb9f80ea390f8035ec8b4fe1ea0707e1bbc80
a924cb1d9651a0e66c673dbc493af84e
443aa64865f504962dd4a50cc115b907487cc175
74b98028f2257d9dc1c95ef2845209f3f585fe6d
abb10f18c87a5b23d21360f80a1ef11c
4ce42b01502c8dd67e0955d185eb840b235f5094
a9845b6596416f0c25e77008dc883794
e8e5c979040094e73dfefc7a7acb1c8c
17f81696b5e3459963cdd483e0efff8027bb5b0b
f898071ea68e0729eb1f6717b0bd899d
8da1744e1b46e4460468621354b5b3edf846bb8a
98613c1522c18d13582c1c1084f21f8dc5bcbd3a
769cdffa9eed1522df6a0004e65e45c0301f29ca
67048d6985baebd06e8aaf774cd2d3b4e9a6a8b4
b71759c62ef96206b72d8d91a1582d24
44e2bae8672a39552e283acb39f27c78a6888535
a804e08ded4328b480c9ecc52fc7acc4
d593c6c4d9d417aca77fbf6a56c39139
6e76b516cfc81072416de5a6f70830f69d281ffe
847c9cdd6d46e1b3b45745b0ab0c3e14
a76bbfbd110e6538192125f2a5940802e16b0c77
f8e1bde5852cbf50fb9ff2c206c18fbd
bae9a086ae39739d661a5694de45b98b
c758ba2525e4a0fce871736829b35023c3d2393a
d713d895e73b2e9a2fcc092d8914c306
50d04338f243c0441111ab81c032d643e667f3e1
8cfdff3aa5ab51f2e71093333c2e9ebe
150d451497e5c1e98c4807bd0d71923e
c4ac2fc8b9f2dcaf8a30cfb5799526269fd3f8a1
05835c4480a9e9fb2e43d99aeefab2c5
c970a11623e406a09a62ed2e07002b0a14fa8ace
e6d8f08694eb06c3c615866e7507ba74
bd5ff7ce5122acc7c06aa4b5ed3a6d385bdd0f8f
5b03744a4371af9527245d549ac7f160
eb56b9d56e882b00bde869bc87307603191c8c6a
6affaa1d4e6c0fba8bb5643c23c103fefc2b37fc
6d0c79e2bf39494563d6422f074bb762
c15a027cae59bdcecb2b8e156c81a6c292f51987
c6c86051e5bc09fa731a81f0f0bb13a2
208dd33cfe69006d7fa40ae0ece37813ab7c6b0d
1b1878f4fa44eb4043903086597a3a5f
603500fd84c97d6301f000f91ab476b6dd122874
4e13683a697f22c512b78cac31a0be7d
44e8a3107b993e98359533b60a92bc57792e241a
ab8505b169095d14c14d66e51b024645
084b35ea7d3941d29bcde88abe820852
ced4f210166fd96095edc7b510b656c1283af9e2
a35e8409033c4e8ca810239f9e670ce6
8b151afe8339c205c03e0a53dcfb73fb178c13f9
ed484f889246dc83200632179315e31d
eed3310f548109a77cce3c768fe10b6a26079025
a162b58efd4d177c4fd28a70c9c10ab8
34395a2b6ff28778dc4d243a8cbf7fd6387f0735
69188b1ddd4ecafd75e2ec4883031590
7c1f56ae4a20bc6f8ebbc7bf9878001a011ec59c
59d166aaf0968cfb640f9d12da4e6bfd
1722f6947bd2d0d2fc6399afe967baa78ec08624
1a2e84e35c8ebc36bc1379192e30c02a
135ceb4b44a8db87f6b246cf33fe44f92a11d142
29625c607ccc2cde669b3a5ea003e3e2
213b9ee92f4a444bd4e0ba91c8bfb64650a7009a
1a0c7dd9fda6d952064ba0d7cf0a3d06
760e67cffaabb2475352c7b479bf70386eb4b9b5
0473a64c89081ba93c93807c7440f5b0
c5c765aae9c9636f365a029dd637927cbc4b356c
5eaea1efa5a6f02be4c4ee4c930eefb7
d4c243ce2f9b618f1f0dea708a1b6d83e520dc31
3335863012ef5d7b4d3a48854992c8b5
9b05711122b45ef9c614d14a4948f369
78069515d9348519238bf4b8dca625866b942d9c
43c4727ce41583b1058ab9027229ab83
70873bdd42b248ff9513c72fc4304c9fe1ef8554
335a8ec208bca6d370a75eb4341df078
7ab60758a16a5d6d1f3855c4cae5d1c9
8ff78422f9983fb83f54edd1a72567aa05a1efdf
d46593fb9009f0fea2639446c96750bb
2d35b7bdf72a4346d0423016079d1c4abb8b95be
d335478695228df139a6b3521465d157
f6ff1b1b6e2e55f54d1de10397c85d48
d0449047bfe79277f38538bed45863b9fe9521f4
edb456461464d967b321ed07fe82b3e2
e63f3a5837a0817169af3601be34202ce757c954
d096d77eabddbbeace4e8be83eba581d
636a0c7a358db2c9b0b1cd5930a040acb2273a02
9f6efe7a023aa1e27e6aa16db1b6dec1
63294a95e0868031c6451a085f403e3c9b776c74
4c09023ee24fae681ee5301bbcf5c672
4e6bcec1e71f4eb614cb92fc7f6fa6352746bc4c
c52ee704e8357e005a66e2b983a8080d
2b779575511590d0bb6b9ee0c45605a8be4f7651
3660a3ffb7bc5fb328c607e62d5866c2
8b109a8c74fa7e42220984f959ce4618
f59f3efeb304086b73372c32d4d5498314b8356e
2a697936033ec5b5ab6db22a1e0a77d2
49b1a985a937a2fe116ec6d67f0274bfb651883f
e8400cf5d26028dcf1464cb255acc0d9
1f1f5920aec65c4953c3c70e652f775d
e60cb2b131544e1e7088efd5191b95f8
16827379cd1ef52d0b77054a473f76d66731b0c5
d86157b81d946ff1ce5c8eb09124dd3b
c5c28b206c886b382ce03450f58cfa417d4c51f8
873eff18b155e9c707b6f4c7b16b1f5d
6cdab92aeec8d4f68ac8670cc39baab9c56996a7
685c29759414a0d1f78c3ffd13de240b
5a0e0434aab0f3a065229f6c4a894ea4657585c8
3b8aee71c8e4a82d439ed547bedbcdf6
158bf5e1f809e2cb9446a267ebe1e08408353d15
b2880d918de3fd9b4f22221cee32e2be
67472d3706e0152e8be5b961d06a77c1ffc777a0
0196d5d8a7fdfa52d82e13361205eec7b82e0010
6cf7ae46caf3a2b3ae6b77a630e8bbfb
9a73dce6d49567ed776212b7a09d72e4073dc7df
5defe01574c8c9304dff23ae83b2edf8
5ff41920a9dab18d8c489068a33959b2d153e76b
248ebc0963fbd7f68f78efa7e953a6ac
c7ba6ffd2a0f712455f7d94ee5fa3069dc418b48
2f74b475d4cd19c31b38b10d5c1e5648
c5b2c75b55eb36a91471d522bf13d4c12cffa8fb
f9d041592f47c3bf8938d68a6754072d
8f16e35747549a04920d7e09d78a6f717ca2e316
d02dd8a6a3bff9ec8a33372eaab19e6e
e3b8662540f01bd5a522e022745da0804ae74343
46cffabe41ee481381e5290b07587727
81714828961914c04c94c408205707d1238f023b
1d61f87b0671a4e8b3e3023c1045e06170f7c442
e124623e49a77120ce5a01202ffe25b7
170f7bdab7b0a44feb98ce7e7e8e6ac7ca81b072
6a06e90686a8da0eea0d6811bf2994e5862ecbef
1afd5b030482168b8a4e954dcf5b43d006f5b6e3
dff9a008fca4df7fa9bfa87e798b986b
c258b758e7e1442bd6065be896a611ae2635a0c8
a871db8f1837bc835e12983b3e9a864a
b5c5a895cb07037bb68e534ae4cb8e332165c20b
186c3731d1dea544ce2b8be28121210f
fd48f4b2e825bc48f22b60c40c5bb548bc968360
77a33cea1f3dfa8db9276f5d11053e9c
73fd6affb33f319b4f49da1c7c911213de3b0976
a0d9941bffffca07cc66a8a5759da1c4
f77281de73af01c67fc1da297254134640806130
9ca37e0ee5a762a53f9f3d9725ce236b
9ee0d9b3470cfb9303de40352a4d2448b31b4bcf
5041d125712609a2155756081edb3edd
f9b0b7d4ceea2a1f2688c76e47f6be09
9ca43aeb90bfdc8923dff7173284d73cc8d69ba3
47eb31b90eb7beec291c62ea2bf59032
ff5e865ac9891e600538da85c50628aae0d6dd0f
0afe89dcbc5b91814d0e8df2c5808c1b3400b618
24cd8d53630714769c90fe016e23a0f0
16121e4f5d9d5335950d056bcbc61866d9284906
5b906be32fda7980b2ffe1e5f49d2caa
2b4f42e80550c1d9e0c0eeee22dfb9eb6b0cbc52
81335b752f7cf0887655d91d3b50b50c
e9d941b0d4620966ff25d96010d5ee1be1f1ff81
e02174be0a8e6daa09ff1d644c9344c1
cba36b07fb1ca31c0ea369aab6dda6df06b7e08b
56defbb2ca6f5a6973f5b81be177da09
e0e4df91a56bc71b3eceaf58e4c9a8c218b1c0eb
d9d690a42c4d0d1862b4fb195ff28f9f
a6621e98bdbabdf48bd9584805cb262184ea6f10
02800134a220e2c6fea1013a53527475
200c076da51366bd6f117172dc2cb850f58d16a8
3ddae78684b51611f096169f031ea57e
4aa3ee7c8827629e9afe93e07b3c3192a7174ed8
92bf17fcc381e85812b0469b6470b126
ea417c25859e354143a32e6861e106e1de1a64e5
a036e9a499df15fcb994f9c93d91dd9c
ab1d343a711d9313fdcdf93dfa03f8c2619f670e
a1b06b2a672d9e7d0fbde0382258fbda
6e7623c226040b68eed85428f6b9906076f57341
e6cb61ac4eb05850b903c795f95cce06
10ee5d556ac289935865053d1dc7dcba6fd86e84
193ec412d0817a46f133c161e0543fe7
4bff5d7660cfc3b08f96bb44fd70c1e565e5d431
666e2361bd30629cf370c6dc9aa4da95
0ca5ccf58a52f6f3ee8bba1a53c063f14655e3ae
e9e93fd4ea348463688fb06a878aa33c
c2a6cf72a870dacff5d2b5e44ca5f7934eb0b88d
7903084e7f3dcb4d10cf9ceb8be43b25
69a5009a2a4ce40c9a8ab7d02b56b9ecf10fcd6d
1a1f421e9c7bc1aa52e7a9e5c4bf13e0
f8e0b97307952a53024f566f31e0b0c9df61f614
20974c9207b915bd6b52a34d11f1678c
b9a247becc4084d0d4d88213d7c47dd6
e272104d3c47e841569137b60d0e7e8eb081a679
fce677cc5bae9e49474464f290ce213045b3d2fe
8fec26aaca38c5892020fbf90899df19
e0dc97a63b048b0d424d48a443fb3616
508835bc6107df70b0ecd0eac30a3faa41830b05
4c34890fb9102a29e5f84c123e69b578
0a94805c682f9bbd719605c2f22c51ecdd48a594
76d47cbf88fd9372192f4ed546de2615
205fa96c604535321da68fc8115bace4c5eca10c
e9572e1ee29da7be229960268718d2e4
5f3f9545ea9bd345a7af9bbeed2c20156ffd2467
a99b3a07e789e001c12aba7ecd59b30d
b3a2ce11edf8234845010dc1b2f27e1b2c590db7
fa96835144b56298de707714799bdd64
2d29c897d49619b529b997b820b92a32
4abb708ba1e66a0029eac33dee4a3edfee2a99c1
bab9635b5348c87e432cb27c0825ce96
6fa9bb915e1bc57358e873324558cd54a3dc9fdd
70d869639fb4b69ee0315d8102fa483fb6f77e00
a5f596b623a41d423f02d708fd8d328a
dfdf521629a0c74f17797bcb4cc8bb56c6f3bb59
2f963389b3376928f37104ef3330f53b
e08fcfca3e93c239083c63eb507b6dbd95acf755
ee0d9d9508269d0dd9ccf3a2ef764954
5534272d096410f5e21bce1a829e1c2f6d660f7a
890c7789d9eb910739bb5feac7e524bc
ab69e7753af6b69fc58ca3960f50da5ee4fac788
a783a6fb8b1e155250a2b79d1b0edf82
aaa278dd04892fec2956b982a746021f
506d656598354c9e0508679f830c748f43b6d34a
732616f521dec8bf2af7105dbab57a0d
b5e42989089d58f27e8f5a05406c199fed6cd447
ee548ac80cbe8006e7926d3159012be0
b5be2a5698b5cf7d667db554511f1a6778e9f73a
19cb709f7c7a15963668af2958f7ffca
06be70629cedde7bb6afb92173ffb71584df1292
ba369f59ac3f3ed96306abf3dc7f0d47
9c87578019849cb3d659f1ebe7e03bb21e80ee57
c9ae9044376533dd0f6659c95351c5ba
fd4ebd29c3f1a5440a49a724ae8e3443
89f653510b8bdb6a4829bc865522e62d
a26ef53133105e81f40cf968f0933f423cd62871
6cb71336fe44bbca936484aab61c5876
9757861d2a967326e003bda54c67fbc31ea57a48
56f299158ec8e1d1500b77e75b2a1e0d
65179490724a1f4d9e0b8b1db144f25b6fec2aa0
9d7ed34f617d9396f799044d22e6622f
4c31b647afdf7a105c6dab22c7a96488de544a7c
f6733d04ec1ee21d4bab8f21a04abd2c
d2e458f4e7bb7d3b7e3a4391eb0f1f1f21a3362f
4308186250f5a43dbda1d3a0bdcce500c6f65676
e5b1956db9f705721d8929e4510a9b05
9c8d0a83904b892c6b0c507d8f747945a1f12b1f
a3753c15ee8b4a5d714ced696b63a012
b8e1560d2ba18f0ae1334639831c2156db1a9b0e
b4814bd11dfe0c9ed749e5707ecaaf2a
0fa30d538620ca1de540db6de407daf1b9f4c958
8321a4171bb6195cd6ef60279d16354e7629b447
bd0ce96b6cb9ce550320921c64b0ae46
9503c70b6795928df4cc07bdfbceab5aab279840
aea7dcaf39654c86a6a005b972be0cc6
7f5668b5b04c1c26751cb0b5953a1dabd3ac3d2a
6fc61c450b59c08d95b8ac732b09baf9
31f6c6997429c144dc1833b8f6a0ef55a8c4601c
f8460dd7954a7342423998f6f9584348
e6f641d1d165bf94702c5ce5a1466958564d6116
a19a0e415984f97e106bf365138f5f9e
7df7a97779235df0bb15ce3b78c1ba68
6f87b10914d8e4af96f611701a398d13ed5dfd50
cb577638f3aaaf09b388b3f156eaaebf
6587041558d3b567be6afd151e22768da558cefe
41c4d9ac95c50c07d7575102b2c321df
cf41323e4a33f056385ee147ee28a6f98ca71e28
3ed3fc84ab54579100b0d564d4d27435
bf1176399dd2c5a2f82a40f14bacf0ff6da60b1c
5652eae66cd40dcc2fd90eb8561307e6
717680bea5d3727ba4f47154415f67043d2eec5f
6a0729f955c69565dd32c3548c94092c
876ce0e9f43c8c73568335dfc34e3291af6fa420
2a28dfc356f957e8904cdf7fc79584e5
e054813ff13ff08161647df953bd5f95
8cf79ac3e9d4d3f8e47e06f0d39df02035b3d326
977588a5c62ffa110b35f664bb62ed55
ebae99450f077ab1f92a1d7ba6eb5443f6280fb2
2d813f21ccd981fd46480a4db867110d
759ee2733e8f76e06c4cf2783443141658741b8d
8c33abccde1472832516076803a437be1076fbf7
a3e56e4a6021301035e9d7d83102eace
91ffca7bc565dae5eeafca09d24114044eb79441
15fe2ff946bcc0efe866c6f35b2aaedf
27d7d540829c899268ab1c05b6d4daeb002436a0
9f23f9c9328fe4afe3efc096df961bf6
1cab661e729c536c797b12810c049ccdf4fb7201
e9cafe44a6a7725a2f4327cd8a58e02f
d470d44d99a6eab669971d60876f3aabc7cc7a0c
f13d29302259e5f3a1975f90f6ac6984
b0d897156088ba3636a27c1eff86f9356d618d3e
69dc87c9491b8e3a460ce44ea4824c87
77b006f9d99db0b1676c2f0db4889e39281764d7
378d8db1f597d9e54b6eb31c46850a5f
1b6c5045f2a74534d69ebbc5c7eebb2ee133805b
097593f1f64b7b820a396196056d5403
b8d5178ca7f22027544cabd8b8ba8b398196885b
aee944a4d2fd8680f97e63d43ff11531
6a7e694a9e5741348281a533c67c22fe0d271dc8
6fd164682c7fcd97a6f71cfe31e003eb
74c9c223f16d23f8fb231af6a60483d65c5bc9f1
48a3ce2d30802c9167e894cc28b1fc62
a4520ddde7028ea1370dda59fc448d2464f8461d
17e71f4f555dc310ec492b823db1688b
144cffca6e44cea9d456c5719ae38625
41763d8bd447dc94d5771732ba8f060aa279caa5
58392518748dd9b96102afcb0a727a1c
a2431bc833a794c69e53d5eace29ca962d654de0
715da5719f86f83b21ba2ef4e5690c476f7251cc
94cc20a3f3262006acb408cf41aa0204
d0760e1d1c0f45218a8b8e7e9e27358fc5dd5a2f
d83be3fba9d7665f97a1cd935f26aecc
b806d78ac75387f111e949af8054267347d4fbd8