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

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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.

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

Permanent Link: http://ufdc.ufl.edu/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


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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.









TABLE OF CONTENTS


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

follow immediately. O

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


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2000.

[2] R. Arratia. On the Stanley-Wilf conjecture for the number of permutations avoiding a
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[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
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[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-,
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[7] M. B6na. On a balanced property of derangements. Electronic Journal of Combina-
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[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;,.:.
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[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,
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[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
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[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
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[15] Micah Coleman. An answer to a question by Wilf on packing distinct patterns in a
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[17] M. Fekete. Uber die Verteilung der Wurzeln bei gewissen algebraischen gleichungen
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[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.,
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[26] A. M. Odlyzko. Handbook of Combinatorics, volume 2, chapter Asymptotic
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[27] A. Price. Packing Densities of L'n r Patterns. PhD thesis, University of
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[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.

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[32] N.J.A. Sloane. The on-line encyclopedia of integer sequences. http://www. research.
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[34] R. Stanley. Enumerative Combinatorics, Volume 2. Cambridge University Press,
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[35] J. Taylor. An Introduction to Measure and Pi, ..',,7.~li Springer, New York, NY,
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[36] V. Vatter. Permutations avoiding two patterns of length three. Electronic Journal of
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[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

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2

PAGE 3

3

PAGE 4

MydeepestloveandadmirationtomywifeHiroko.Whilestudyinginaforeignlanguageinaforeignland,shetookuptwoofthehardestimaginableroles,thatofmilitaryspouseandthatofmathematiciancaretaker.Daisuki!IthankmyparentsBobandBobbiColeman,mybrotherMatt,andmivecinoAbbyfortheirpatience,humor,andintegrity.GreatthanksgotoProfessorsJulieMiller,TinaCarter,andNormLevin,forrstintroducingmeto\realmath",toourGraduateCoordinatorPaulRobinson,andtothegreatestadvisorycommitteeeverassembled,ProfessorsDavidDrake,MeeraSitharam,AndrewVince,andNeilWhite.Iamhonoredandhumbledtobeassociatedwitheachofthem.Finally,mydeepestrespectandappreciationareheldformyadvisor,BonaMiklos. 4

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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

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....................................... 88 BIOGRAPHICALSKETCH ................................ 91 6

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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

31 ]launchedpatternavoidanceandcontainedsomeresultswhicharestillhallmarksoftheeld.Asthenamewouldimply,enumerativecombinatorialistsmostenjoyenumeratingsets,thatis,determiningapreciseformulaforthecardinalityofeachsetwhichdependsonlyontheindexorindicesofthatset.Unfortunately,weoftenndquiteinterestingdiscreteobjectswhosenatureiscomplexenoughtoeludepreciseformulae.Alas,wewillseethatformostpatterns,thesequencesn()fallsintothelattercategory.However,allisnotlost.Aswasbrieydiscussedintheintroductorychapter,greatinformationcanstillbehadbytheasymptoticsofasequence,andmanyofthecurrentresultsintheeldofpermutationpatternsinvolveboundsandlimitswhicharenotasstrongaspreciseformulae,butcarrypowerandbeautyoftheirown.Letusrstseesomeexamplesofpatternsforwhichwecangiveanexactformula.WewilltreatallpatternsinSmfor0m3.WewillmakeuseoftheKroneckerdeltai;j,denedbyi;j=8>><>>:1ifi=j;0ifi6=j:

PAGE 17

Thereadermay(should)havebeenamazedbythefactthatsn(12)=sn(21)andtheirdualproofs.Infact,thedualityinvolvedwasthat12and21arereversesofeachother,andforeachntheuniquepermutationinSn(12),themonotonedecreasingpermutation,isthereverseofthemonotoneincreasingpermutation,theuniqueelementofSn(21).Ofcourse,onecouldalsoprovetheequalitywiththefactthat12and21arecomplementsofeachother.Perhapsthesestatementsalsoapplytolonger,moreinterestingpatterns? 17

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18

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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.

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31 ])Foralln0,sn(123)=sn(132): 20

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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

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21 ].Foraclearandconcisetreatmentofall,seesection4.5of[ 5 ].ThereaderisalsoencouragedtoseeDoronZeilberger'salternativerendition[ 41 ].ThereisstillhopeforatighterproofoftheStanley-WilfConjecture,astheFuredi-HajnalConjectureonlyprovesthereexistsaconstant,buttheconstantswhichwegetfromtheproofareastronomicallylargerthantheobservedconstants.Itdoesstillgivestructuretoourworktoknowthatforanypattern,sn()isatmostexponential,andadditionallythelimitlimn!1sn()1=nexists.WedenotethislimitbyL()asitiscriticaltothesequel.Now,foranitesetofatleasttwopatterns,wedonothavesuchastronggeneralresult.Foranysuchsetofpatterns,itisreadilyseenthatsn()
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Proof. 17 ]forsuperadditivesequences.TheanalogofthislemmaforsubadditivesequenceswasusedinArratia'sproofforthecaseofasinglepattern. Withthesefactsinmind,wedeneLonsetsoftwopatternsasfollows. 23

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3 ,thelimitandlimsupagree,andLisaswewantittobe. 2-2 .Notethatinourpreviousexampleeachentryofthebackendislessthanthethreshold.Creatingourowngoodluck,wechose532614forourexamplespecicallybecauseitavoids123.Infact,every123-avoidingpermutationsharesthisproperty,asimplestructureofwhichweshalltakegreatadvantageinourhandlingofthesepermutations.Toseethisproperty,supposeourthresholdistandthereisanentryjinthebackend(equivalentlyj>t)withj>t.Bydenitionofascendee,t1
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Thepermutation532614.

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39 ]JulianWestdenesageneratingtreeasarooted,labeledtreehavingthepropertythatthelabelsofthechildrenofeachnodexcanbedeterminedfromthelabelofxitself.Thisleadstothecharacterizationofageneratingtreebythelabelofitsrootand 26

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Thepermutation865321947. asetofsuccessionruleswhichdeterminethenumberofchildrenandlabelsofchildrenforeachnodeofagivenlengthandlabel.Theclassictaskforacombinatorialenumerologististodeterminethenumberofsomecombinatorialobjectsofsizen,perhapsfurtherindexedwithrespecttosomepropertyorsomestatistick.Typically,oneispresentedwithaninitialobjectofsomesmallsizeandarecursionrulewhichsayshowmanyobjectsofeachsuccessivegeneration(objectsofsizen+1)canbecreatedinductivelyfromthoseofthepreviousgeneration(objectsofsizen).Denethegthlevel-numberofatreetobethenumberofnodesinthegthgeneration.Thusthegeneratingtreeiseasilyseenasatoolwhichlendsitselfquitereadilytocombinatorialenumeration.Weconsiderthenodesofourtreetobethecombinatorialobjectsthemselves.Therearemanysituationswhenthenumberof(n+1)-objectswhich 27

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QQQQQQQQQ Arootedtree. canbegeneratedfromanyn-objectisallweneedtoknow,sowemightaswelllabeleachnodewithitsdepth.In[ 39 ]Westbeginswithatrivialexample,thecompletebinarytree.Webeginwitharootwithlabel(2).Oursuccessionruleisthateachnodewithlabel(2)hastwochildrenalsolabeled(2). 39 ],Example1). 28

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Thecompletebinarytree. 39 ],Example3).

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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

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2-7 .Anactivesiteinapermutationisavalidinsertionpoint,thatis,asitewherewecaninsertn+1andobtainachildwhichisstillinthecurrentgeneratingtree,soforourpurposesanactivesiteissuchthattheinsertionwillnotcauseanoccurenceofanypatternwhichweseektoavoid.Thedepthofapermutationisthenumberofactivesitesin,equivalenttothenotionofdepthdenedaboveongeneratingtrees.Wenotethatthedepthdependsonboththepermutationitselfandonthetree,specicallythepatternbeingavoidedwhichdeterminesthetree. 31

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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.

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4 ].

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Proof. Proof. 34

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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

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Proof. Withthislemmainhand,wesubtlyalteranotherproofofBonatoachieve: 6 thereareatmost(r1)2(k1)possiblepermutations 36

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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

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Proof. 2-8 39

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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

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5 ]Denition5.33)Apermutationiscalledlayeredifitcanbewrittenastheconcatenationq1q2qkwhereeachqiisadecreasingsequenceofconsecutiveintegersandtheleadingentryofqiissmallerthantheleadingentryofqi+1for1ik1. 2-9 Thelayeredpermutation213654withlayers21,3,and654 Manyresultsareknownconcerningpatternavoidanceandpatternpackingforlayeredpermutations.OnecanseeSection5.2.2of[ 5 ],[ 27 ],[ 6 ],and[ 20 ].Ournextresultisonlayeredpatternswithjusttwolayers,equivalentlynon-monotonelayeredpatternswhichavoid123. 41

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42

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23 .Forexample,foranyd>2,wehavethe(123,3142)-avoidingpermutation(d+1)(d1)(d2)1(d+2)d,whichhasdepthd.Thismeanswecanarenotguaranteedaxedboundtothedegreesofourgeneratingpolynomialsofthelevel-numbersofourtrees.Allisnotlost,however,aswecanshowthatforanyofaclassofpatternsandanykthereisanupperboundtothenumberoftreesinWwhoseroothasdepthwhichisk

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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:

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2: 45

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2followsimmediatelyfromtheabovediscussionoftheasymptoticsoftheFibonaccinumbers. 2: 46

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47

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48

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Proof. 49

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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

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9 ,weknowthatthefrontendof(0)hasatmostmmaximaldecreasingintervals.Infact,thereareexactlyMmaximaldecreasingintervalsinthesubsequencer+1t1forsomeM
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Inthepreviouschapterwediscussedequivalenceclassesforpatternavoidance.Ifthepermutationcontainsthepattern,then1contains1.SimilarlyforRandC.Fromthisweimmediatelyachieveforallpermutationspat()=pat(1)=pat(R)=pat(C):So,;1;R;andCareinthesameequivalenceclasswithrespecttothefunctionpat.Next,wedenethefunctionmaxpatoverallnonnegativeintegersbymaxpat(n)=maxfpat()j2Sng:

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3-1 foranexample.Theclaimisthatforalln1,pat(W(n))Fn,whereFnisthenthFibonaccinumber,asdenedintheprevioussection. 2liminfn!1maxpat(n)1=nlimsupn!1maxpat(n)1=n2:

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ThepermutationW(6)=342516

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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

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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

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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:

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60

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Itmaybenotedthattheseclassesofpermutationsresembletheclassesusedin[ 15 ],[ 16 ],[ 14 ],and[ 25 ],exceptthatwhereasineachofthesepaperstheconstructionconsistedofkrowsofkentrieseach,orastrippeddownversionofthat,thepermutationsweareusingherehaverestrictedrowlengths(numberofmaximaldecreasingsubsequences)duetotheq-avoidingrestriction.Itwasnotedthatk!1.Fromthisfactweseethatasweletr!1,ourconstructionsforincreasingpatternsoflengthr+1\approach"theconstructionsforthegeneralcaseandtheliminfsapproach2. 61

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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

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18 ],Taylor[ 35 ],andChung[ 13 ].Fora 63

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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

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35 ]. 65

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7 ],[ 9 ]).

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7 ]and[ 9 ] q,aprimitiveqthrootofunity. Proof. 4{5 )isageometricsum,andwetakeadvantageofthepropertythatthe(kr)-rootsofunityaddupto1toseethatq1Xt=0(kr)t=8>><>>:0ifkr6=1:qifkr=1:

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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()

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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:

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3 ]whichfollowsfromTheorem 39 : 41 andkeepingtheabovenotation,wehaveshownthefollowing.

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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

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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

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x+1;sowelet2=1cos x+1forallx2[;1].Finally,set=minf1;2;3g. 21 11 andTheorem 42 ,thesignlessStirlingnumbersoftherstkindareasymptoticallynormalandthusuniform.

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74

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5{2 )isan;k.WeapplythemethodofInclusion-Exclusion.Thereareknwaystodistributetheballstotheboxes,aswehavekchoicesofwhattodowitheachofnballs.However,wemayhaveovercounteddistributionsinwhichaboxwasleftempty,sowesubtractthek1(k1)nwaystochooseaboxtobeemptyanddistributetheballstotheremainingk1boxes.Now,wehaveovercountedthedistributionsinwhichtherearetwoemptyboxes.Indeed,forsome1p
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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

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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

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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

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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.

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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

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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:

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k1n1 k1nFk(n): 22 .Weprovedthestatementforn;2andn;3inLemmas 21 and 22 ,respectively.Letk4andassumebyinductionthatthestatementholdsforalln;jwith2jknk:

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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

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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
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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

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5{9 )ofLemma 20 as (k1)2nG(n);whereG(x)=F3(x)Fk2(x)F2k(x).Set=k2(k2) (k1)2andnotethatk1<
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5{15 )by(k1)n1 k1=k2(k2) (k1)3.Nowitsucestoannihilatealltermsonthelefthandsideexceptn;k.ByTheorem 48 ,k+1 87

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[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

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[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.

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[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.

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AnativeFloridaCracker,IgraduatedfromSeabreezeHighSchoolin1994andenteredcollegeasamusicmajor.Realizingmylackoftheskillanddevotionnecessaryforaprofessionalmusician,IenlistedintheUnitedStatesNavy,stationedforfouryearsinYokosuka,Japan,whereImetmywifeHirokoShinohara.Aftermyactivedutyperiod,IcompletedanAssociate'sdegreeatDaytonaBeachCommunityCollegeandjoinedmybrotherandhiswifehereinGainesville,enteringtheworldofhighermathforthersttimeattheUniversityofFlorida.AsajuniorIrstmetmyadvisorandfellinlovewithCombinatorics.Igraduatedsummacumlaudein2004andacceptedafellowshipatUF.Ihavethoroughlyenjoyedmygraduatecareer,evenmyeight-monthsabbaticaltoBaghdad,Iraqlastyear,fullyfundedbytheNavalReserve.IlookforwardtoalongcareerofmathematicsresearchandhopetomakesomesmallcontributiontotheConversation. 91


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1191 F20110331_AADEQE coleman_m_Page_25.txt
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1318 F20110331_AADEQH coleman_m_Page_29.txt
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1251 F20110331_AADEQI coleman_m_Page_30.txt
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1179 F20110331_AADEQJ coleman_m_Page_32.txt
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1577 F20110331_AADEPV coleman_m_Page_11.txt
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2001 F20110331_AADEQK coleman_m_Page_34.txt
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2177 F20110331_AADEPW coleman_m_Page_12.txt
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1790 F20110331_AADEQL coleman_m_Page_35.txt
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1696 F20110331_AADEPX coleman_m_Page_14.txt
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1132 F20110331_AADERA coleman_m_Page_55.txt
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2053 F20110331_AADEQM coleman_m_Page_37.txt
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1849 F20110331_AADEPY coleman_m_Page_15.txt
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1750 F20110331_AADERB coleman_m_Page_56.txt
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1860 F20110331_AADEQN coleman_m_Page_38.txt
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1671 F20110331_AADERC coleman_m_Page_58.txt
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1740 F20110331_AADEQO coleman_m_Page_39.txt
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1907 F20110331_AADEPZ coleman_m_Page_16.txt
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1763 F20110331_AADERD coleman_m_Page_59.txt
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1108 F20110331_AADEQP coleman_m_Page_40.txt
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2073 F20110331_AADERE coleman_m_Page_61.txt
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1519 F20110331_AADEQQ coleman_m_Page_42.txt
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1560 F20110331_AADERF coleman_m_Page_62.txt
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1707 F20110331_AADEQR coleman_m_Page_43.txt
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2023 F20110331_AADERG coleman_m_Page_63.txt
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1277 F20110331_AADEQS coleman_m_Page_44.txt
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1746 F20110331_AADERH coleman_m_Page_65.txt
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1815 F20110331_AADEQT coleman_m_Page_45.txt
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1906 F20110331_AADERI coleman_m_Page_66.txt
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1821 F20110331_AADEQU coleman_m_Page_46.txt
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1909 F20110331_AADERJ coleman_m_Page_67.txt
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2f1517c81d494f7d800fdf011f2e1125d3ee5182
1797 F20110331_AADERK coleman_m_Page_68.txt
4840c93b9f7ef155ed1b1e235ee83ef9
efd419a9914bee49beb2c74e029c7fe044b6e7cb
2381 F20110331_AADEQV coleman_m_Page_47.txt
eab0af2c15c59df69f4778b0dccb565e
a8cb494fd20a7a6856cda57a6607d3c532542540
1749 F20110331_AADERL coleman_m_Page_69.txt
5d1e8c15382bd2265abcedd0c7f34e93
03005f9f26248c0cce8277c1e2349833dbc718b2
1630 F20110331_AADEQW coleman_m_Page_48.txt
8ce9a291fa6652edefbe8ad5a1aa1482
90406e8bf98e1ada6a3704347645cd3d759a1e15
1692 F20110331_AADERM coleman_m_Page_70.txt
9fd595b5469a18a0b614168f82e5ce0a
183d6b386e74d7169210b67b6f0e05308479c36b
2170 F20110331_AADEQX coleman_m_Page_49.txt
49d01e772dc1b2c8d18092f8ed0ebbd0
690e79f9d7953f97ad66c404a555f67f50fc59f8
1059 F20110331_AADESA coleman_m_Page_91.txt
43778066a334ccdbd7504d2d8590c9e5
8a5d49e821e35b3e5265308524411d2b410a4d9f
2070 F20110331_AADERN coleman_m_Page_74.txt
f5353e8562f3ba9eb05d38dab806cffd
2781fa895ff70db493a26e265543702ae46af4fc
538 F20110331_AADEQY coleman_m_Page_52.txt
8b72071dca7a21e7f409322de9784ab7
373a797879392b4fe4d719db033c61f7da0952f8
6972 F20110331_AADESB coleman_m_Page_03.pro
cb0719dbdaec587fc5f5a939cbee045e
2c2aa682c88b6fac64b1e8368ab8ec0b9817bd72
1992 F20110331_AADERO coleman_m_Page_75.txt
b49e3436cd7554ce369986826b41b810
8aa6132fd7c3587657a0a54d48c7a94da16620ef
1597 F20110331_AADEQZ coleman_m_Page_54.txt
1d4707539c7f559994dee4035aa6a1da
6fe6cb694f60bfc36b1605657b3d0fefb51d6d99
20873 F20110331_AADESC coleman_m_Page_04.pro
e92b58d60a0b33698d219b405bf323c2
648c706d105c35a9b6a58e90c30c32f55585d830
1423 F20110331_AADERP coleman_m_Page_76.txt
226ef9845442d7973c3b52c5084bd836
806b3edd319c2ab2fb506169cb5c6412e467bd3d
63460 F20110331_AADESD coleman_m_Page_05.pro
ec6fd3acfe71c1171e2a7a731b55e7fe
9ac69291e8bbdd618ebc2e67996cf4dec2aaefdd
1263 F20110331_AADERQ coleman_m_Page_77.txt
588869665d651f0ddcad276a703ef936
88e4ae21ed12565b077276eec9f675b240d44a34
13088 F20110331_AADESE coleman_m_Page_07.pro
6dc55a7970ae47ed4bf7729a8f8e25fe
e7731e799720122d9a4d01ac7ca7ced08741d0ae
1644 F20110331_AADERR coleman_m_Page_78.txt
83c9090610887cfb416b285d80aad6e3
0ef016e515dc2d22807b41bb1c02f8265025b5a1
10170 F20110331_AADESF coleman_m_Page_09.pro
21683fcf68719b981c21879072bdfa4c
6e01612df135112363059cbfc9628f7fa927cb74
1769 F20110331_AADERS coleman_m_Page_79.txt
f1514acf3f5b477a2a1d1f97e716766b
81a7c3cb111d6815dabde3e1e6c72c821ccf7b84
44891 F20110331_AADESG coleman_m_Page_10.pro
a824f7da3fb92f0f7292f63f7444bca0
44335f0aa915852ff97053d3376472fc62fdd0ef
1656 F20110331_AADERT coleman_m_Page_80.txt
1ae0c07ab55f8043041d5552553fd865
e089f01a6c892316774eb05afc8e29908c8bbc8c
53885 F20110331_AADESH coleman_m_Page_12.pro
dd48647f6f72f4ae178ac43e65098b43
d209567d82a718bbf198c72d54f804d9e6d40390
1286 F20110331_AADERU coleman_m_Page_81.txt
7852b1d8e305be29e70007bbcdce4e01
9160d9733cc23c2eef49c9838e4c46b7566cbbf4
28205 F20110331_AADESI coleman_m_Page_13.pro
762a67f101141c7bca564e4713d7099e
becf0087b556e7fa550f9a91e45480e487dd93be
1999 F20110331_AADERV coleman_m_Page_86.txt
d90b778b2138cb8d9523c3dc987d43c4
68e310350aa1efef0383170da86b243e181fa896
39156 F20110331_AADESJ coleman_m_Page_14.pro
0a4afba84f3e7956cb658eaa1aa20683
20ff62d6be3c055037f7336dd69bc7f4840d7abd
43889 F20110331_AADESK coleman_m_Page_15.pro
10232405c1b2e8b6d06e1d2832b24192
e37bf20492202454ed0a72cdc21d3cca03bf3990
705 F20110331_AADERW coleman_m_Page_87.txt
6b0a33b77f71d04bc14c9604e89af887
3af23e99d0a77e229ea25498052b1be74b12ad18
34329 F20110331_AADESL coleman_m_Page_17.pro
808193e046bb12b2ebd2af8812113c51
c34c7b5731694f7ded0a7e2e6d151febd592af24
2364 F20110331_AADERX coleman_m_Page_88.txt
1a2ede87bf8ef3b3cb735396089e6a0c
1f238e32871ea53aab6ea2b55c8b3a066704ccc7
38901 F20110331_AADETA coleman_m_Page_35.pro
d6daa527d953e0eb1b253f952d8d3950
6c41a549d7de02d71be4ad8333dda63b5940369f
29138 F20110331_AADESM coleman_m_Page_18.pro
d96167428b7500348ffe0cf641bf02ec
6140d7a44e2e031f7cb1d7c6bed9a8c37d18a526
2216 F20110331_AADERY coleman_m_Page_89.txt
27b134dd54cd2b80acb56cf98737f640
cac31ae236463717d7e800579ab45341ca793adc
46904 F20110331_AADETB coleman_m_Page_37.pro
830353be1bf022204d665a87cf4390c1
66614582f25f3b6700562bed739b3a67b7429fe1
42140 F20110331_AADESN coleman_m_Page_19.pro
8a82475129732d160b4cd4db6838fdc4
f1cdb3fc2bf12c3ae5e6f2b2365236009767edf9
860 F20110331_AADERZ coleman_m_Page_90.txt
d42794a9584bd6827fcebb12193be8fe
6381b0380c25eafe5dc0c96fb1bf559a0343893e
40557 F20110331_AADETC coleman_m_Page_38.pro
2e26d4c11430d9eadfb13d9d632988df
216af00c3ebcbac436ea44263b01c72ee2f45081
40184 F20110331_AADESO coleman_m_Page_20.pro
8454309cb8b6c08b7ae49b002b606ed4
88328c12c0bb9eb7d86bef6bfde2029da72e5a23
20578 F20110331_AADETD coleman_m_Page_40.pro
ec86ff23b9b1b0afd3613b5192d184c1
54feae99a41db96b057a89121ff29d1a9a3145b2
43063 F20110331_AADESP coleman_m_Page_21.pro
56fa6fce3a5d75d6af50b20561f7576b
4d788fb725c34ed5996eae51a0bf92237eb1badf
22564 F20110331_AADETE coleman_m_Page_41.pro
d5635c967eda848f433ad61e6e9715bf
db9d0a904fb1d911c2106e0e0f24038afad9052a
32436 F20110331_AADESQ coleman_m_Page_23.pro
173052e5596558a23786f9abc726c74a
55e46b0ec0010dae8c4036c4fa49704da0393100
37200 F20110331_AADETF coleman_m_Page_42.pro
553bcdb1fda96e7bd286c7b194779022
71e7b99ea43b9a40e610b1032e2c0fe3d57bc0cf
51973 F20110331_AADESR coleman_m_Page_24.pro
0bc4d51ce48ee9ebf8aad5ddc8e678aa
6e0533e23519ad20cc6544664d34cac00fa170cb
25978 F20110331_AADETG coleman_m_Page_44.pro
5180e596c5ab1bf625eaaba0a8c4b87a
5bdcc7c05f49ab37ce7625248e48aa0bff55bcf7
27520 F20110331_AADESS coleman_m_Page_25.pro
12f26ddff36876e687780a088cb7b49c
cde6213ce42c2d38b873ce77a18f95b1362fb94a
34871 F20110331_AADETH coleman_m_Page_45.pro
217f27f98fb01f2699f21039ac113cf3
1f5ace740ac5625ec27a58fcd649b8258772400a
26751 F20110331_AADEST coleman_m_Page_27.pro
818ce247acf30055313f8e565bd2a2d6
afb92f833cf021b125b5c5d2930c4e696d2752c4
44734 F20110331_AADETI coleman_m_Page_46.pro
0ed02517d6f31696603d21eb88823f8d
7353246acae99723b65cf4680263a0d79123e0de
25337 F20110331_AADESU coleman_m_Page_28.pro
574a48bf0d0864e199836fb9b9a504a8
1ca2de7f5e871a7e129110423fcc2ed431790251
60673 F20110331_AADETJ coleman_m_Page_47.pro
cb793de95101e49dec510c571477a7ef
f0dfa89f3a0593ccbea7f3d5ff38d817dbfffefb
24644 F20110331_AADESV coleman_m_Page_29.pro
a9c10c1ce55ca9ad34be17913f5a0013
614437532e9fbc7e9d98fb52d139af606fc3be34
52767 F20110331_AADETK coleman_m_Page_50.pro
53600001bab3c7dbae5bc240394f87a3
084af8c5e27fe51cb8f0840865a1075369158184
29052 F20110331_AADESW coleman_m_Page_30.pro
c79c536ef71a9c92c8afa6c1e6104506
4fd50c863328460441c48e0c6971aea9731f1ae1
32053 F20110331_AADETL coleman_m_Page_54.pro
201f66e46535e93f54505a8a6a2301a4
6ed6d6b7eaad49624335ce4c9ae2766991717c3b
44959 F20110331_AADEUA coleman_m_Page_75.pro
c859d0bb589063b8bc42127db5cc3796
a11a0019d311fbf0830072fc74b23f9996a81799
38627 F20110331_AADETM coleman_m_Page_56.pro
8b83fc9b8d7c354f89975fec652475e0
2ab13a808cae53570e2ec779e84a005de692e5dc
24980 F20110331_AADESX coleman_m_Page_32.pro
5bd70973102aff1e3fd5f561b5edfab5
b582438056701b48f6b5d2672cd1a056b8e78b66
20140 F20110331_AADEUB coleman_m_Page_77.pro
5dfb29f097a277a48f8c01ccadb1007b
7aeb9bcccb498ed6438198676a03f1d2afe2dce0
44997 F20110331_AADETN coleman_m_Page_57.pro
a21ee0627ac880bad78a2de27b1057c5
9c2ca8d31d6954cf1d7958e557d7a27b654cb7a4
34366 F20110331_AADESY coleman_m_Page_33.pro
5d944525fcf7be884b5490b639ce31ed
c004c234be2618fddd91da805d8aeeb6b128371b
37857 F20110331_AADEUC coleman_m_Page_78.pro
6ff354819abef64c55e3fe43a67c7eb5
dcd3e4f2d62ea3fda6f96f2405d22d4eac34f881
34975 F20110331_AADETO coleman_m_Page_58.pro
9541176c6cb468f192fee85c5e6701d0
201c532e2ab84051cc9c7cc5931f5b81203a98cd
49992 F20110331_AADESZ coleman_m_Page_34.pro
d4629877655d04cb3649d92574f87c74
aa90b601b3b2414cee3310f0768bf87a5a7d0f08
34905 F20110331_AADEUD coleman_m_Page_79.pro
852f5821b0af811a48eed0aa5de2886e
e596b79623f7742f4c532511638401b3d105e3b0
34208 F20110331_AADETP coleman_m_Page_60.pro
fc356ca8bcbb05883291c074798f6a80
60dd3e89b946d7b48ef37a4ca9a76df5dcf904b6
17846 F20110331_AADEUE coleman_m_Page_81.pro
dfb29a332143725403311c4c9b9c2901
388f9f54de227df497d6cd4deb84db79eb67a43b
35798 F20110331_AADETQ coleman_m_Page_62.pro
5e9f5ddbe479c9199cbb4404299e54bb
798ed36e52b116713fd51dbb2913d372aaef09a2
28058 F20110331_AADEUF coleman_m_Page_82.pro
e0f5bea2cb3d7c958e0e89c2eae13c90
30ff75226535c58f25e5cd82e3d4198743274fb1
46153 F20110331_AADETR coleman_m_Page_63.pro
2f005d8af111474478479fe20101bf7a
1fc3aa29c37a94ed03f15b68eac3dce86ca5433a
1051966 F20110331_AADFAA coleman_m_Page_36.jp2
f28683c0ecfd08fa1935c92aee28fc7d
3020afc532c0fe1af9336e10a42d6dc2b637fb81
23700 F20110331_AADEUG coleman_m_Page_85.pro
430b2c24aea6e64969334c3954868146
b8a00d7047e8b35c470f2f8dea76430da8907500
43035 F20110331_AADETS coleman_m_Page_64.pro
b27f4920310c79f4b021627d80858ff5
76fb554615982f805de6625eb79d768b39010049
84152 F20110331_AADFAB coleman_m_Page_38.jp2
ba3a6c18069f5a801a9594d019f0a573
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35423 F20110331_AADEUH coleman_m_Page_86.pro
198bc828d9156574509a3c3aa64c5ad3
42bce7b5d30159d7ba859dbb6f0488600ed2f067
37913 F20110331_AADETT coleman_m_Page_65.pro
2d0c1bd6b47fa80a23b1c2b9f59d07e0
20846abdc69d4c915bf338e993f8acac4283590f
888425 F20110331_AADFAC coleman_m_Page_39.jp2
e799e1b9645fcfe685021695a4eff9f4
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59115 F20110331_AADEUI coleman_m_Page_88.pro
2ef434f2f0499d31cca556ceab4139fd
e15c37add655672a8bbcd229f370ac6ae6735948
42389 F20110331_AADETU coleman_m_Page_66.pro
b7719ba7d65af9942ee42c3b8be14ffb
478eaebf026bee5c0bd731607795af7ccc0d572f
49410 F20110331_AADFAD coleman_m_Page_40.jp2
00430ca926525738ca6bdc6a3d9c1d42
fcd3e7321b0530aa25ce94310da62f1ea9bb508e
55310 F20110331_AADEUJ coleman_m_Page_89.pro
47678fdae618282429d3adf42f2ff95d
d232541e1e4b35f6e5e5595c1ac64d42f669c394
30910 F20110331_AADETV coleman_m_Page_67.pro
e32a991dd8ca5e11d70dc38edb1431ac
35a6a3412636ac9ee0872797fce079d648869daf
25828 F20110331_AADEUK coleman_m_Page_91.pro
27caa5e2597cdfee4715492bc7193f59
bbc804ef56370890a88014bbf5da3449edbe95ac
33262 F20110331_AADETW coleman_m_Page_71.pro
3ad786e7ebfefc1605713bd22fb15408
9eeaea8a6eef06feaf8d0e6057ff3504f7bd49df
76906 F20110331_AADFAE coleman_m_Page_42.jp2
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8aa2e5a5bc07bc96a2be4334c13b693a
7da2adf22a51ad08c3f92d711c153d15e39292ee
27435 F20110331_AADETX coleman_m_Page_72.pro
0e0851c03a6f782d9345857ed8df509f
224af84138ef0cab509006dd75f2bfab4a316136
62179 F20110331_AADFAF coleman_m_Page_44.jp2
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73054 F20110331_AADFAG coleman_m_Page_45.jp2
163c2248fb6c91cf33539b3b927f9d2b
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20749 F20110331_AADETY coleman_m_Page_73.pro
c72b577c5c2e74f9211333c022643266
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122692 F20110331_AADFAH coleman_m_Page_47.jp2
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48729 F20110331_AADETZ coleman_m_Page_74.pro
99203c649b7f7de06ad8084d961a037e
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110011 F20110331_AADFAI coleman_m_Page_49.jp2
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