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OPTIMIZING THE PACKING BEHAVIOR OF LAYERED PERMUTATION PATTERNS By DANIEL E. WARREN 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 2005 Copyright 2005 by Daniel E. Warren ACKNOWLEDGMENTS My first and foremost thanks must go to my wife Jennifer, the Mathews and Warren families, and all associated, whose many years of hard work have paid my passage to this point, and without whose lifelong support this research could never have been possible. I would also like to express my appreciation for the hard work of my advisor, Dr. Mikl6s B6na, first in drawing me into this beautiful subject, then in careful guidance along the path to research. My deep gratitude also goes to Dr. Jonathan King, for his many contributions to my personal and professional life over the past two years, and to the members of my advisory committee, Dr. David Drake, Dr. Meera Sitharam, Dr. Andrew Vince, and Dr. Neil White, for guidance along the way, careful reading of this dissertation, and donation of time in general. TABLE OF CONTENTS page ACKNOWLEDGMENTS ........................... iii LIST OF FIGURES ...................... ......... vi INDEX OF NOTATION ................... ........ viii ABSTRACT ...... ............................. x 1 BACKGROUND AND RELATED PROBLEMS ......... .. 1 1.1 Permutations, Occurrences, and L i..rs ................ 1 1.2 Permutations and Posets ........... ............ 4 1.3 Sorting Permutations with Stacks ........ .......... 5 1.4 Pattern Avoidance .................. .. 6 2 PERMUTATION PACKING ................ ... .... 9 2.1 Definitions and Notation .................. ..... 9 2.2 The Impact of L . ii.I. ................ ... .... .. 11 2.3 NearOptimality ............... .......... .. 12 2.4 Expository Examples ............... ....... .. 13 3 PATTERNS OF BOUNDED TYPE ........... ..... .... 19 3.1 Permutations, Partitions, and Complexity . . ..... 19 3.2 Simple and AlmostSimple Patterns ... . . 23 3.3 Increasing Patterns ............... ........ .. 24 3.4 Supersymmetric Patterns .................. .... .. 29 3.5 A Partial Order on the L i, i. Patterns in S . . 32 4 PATTERNS OF UNBOUNDED TYPE. ................. 34 4.1 The L ,.i Pattern 1 .. . .... . . . 34 4.2 L .i rs and Antiliv,f.r .................. ...... .. 35 4.3 The L i, 1. I Pattern . ....................... 36 4.4 The L i 1. t Pattern 2, . . .. . . . 38 4.5 Unimodality and LogConcavity ............ .. .. .. 48 4.6 Application to Larger Patterns ................ .. .. 52 4.7 Concrete Results .................. ......... .. 56 5 EXTENSIONS TO MORE GENERAL PROBLEMS . . 60 5.1 On Uni ,, i t Patterns ................... ...... .. 60 5.2 Packing Sets of Patterns .................. ... .. 63 5.3 Allowing Repeated Letters .................. .. 64 REFERENCES ...... ............. ............... .. 67 BIOGRAPHICAL SKETCH .................. .......... .. 69 LIST OF FIGURES Figure page 11 The set {1, 2, 4, 7} is an occurrence of 2314 in 34215876 . . 2 12 3215467 is the unique 1li 1,. permutation of type (3, 2, 1, 1). Liv.rs are shown in shaded boxes .............. ... 3 13 The poset P, for 7r 1327654. .................. .... 4 14 Swi' hv,rd Stack ............... ..... 5 21 Longer permutations having two lV. ir of equal length correspond to the partition of unity having two equal parts. ........... ..13 22 Structure of a 132optimal permutation. The boxed area is a scaled down copy of the whole permutation. ................ 17 31 Here, r = (2, 3, 4) and A 6 1. The function q(r, A) gives the probability that the first two balls occur in a box Bi, the next three balls occur in a box Bj, and the last four balls occur in a box Bk, for i < j < k.. 21 32 Here again, T = (2, 3, 4) and A 6 1. The formula 3..5 (2, 3) gives the probability that the third, fourth, and fifth balls occur in a box Bj and the last four balls occur in a box Bk for some 3 < j < k < 5. The set of balls and boxes to which we are restricting our attention is shown boxed. .................. .. ...... 25 33 The proposed order on the 1 li I patterns in S4 . . 33 41 (a) An occurrence of the pattern w = 32154 in a 5 4 3 2 1 7 6 10 9 8 is circled. Notice that every livr of 7 is contained in a l vr of a. (b) An occurrence of 7r 12543 in a 2 1 4 3 6 5 10 9 8 7 is circled. See how the .iiI ,liier of 7 may "climb" several rs of a. . .... 36 42 The form of a as a single 8antili vr followed by a list of li.i.s. ..... .40 43 There can be no occurrences of 7,,, inside the boxed area, so it will not destroy any occurrences of 7,,, to switch the positions of La and A. ..... .............. ................. .. 40 44 Lessspecific structure of the permutation a .............. ..45 45 An occurrence To of T2, in an unknown permutation a is shown for y=4. Notice that we can only get an occurrence of T,6,) containing To by adding elements to the shaded regions one to the left region and two to the right. The only assumption we make about a is that it is li . . . . . . .. .. .. 58 51 The first three iterations of the algorithm applied to r = 3142. . 60 INDEX OF NOTATION Here is a list of notation used in this text, in the order that it is introduced. The page numbers correspond to first usages of each symbol. [n]: the set of the first n positive integers ............... .. ................... .. 1 S,: the symmetric group on [n] ..................................... ...... 1 7rev: the reverse permutation of .......................................... 1 t": the complementary permutation of 7r ..................................... 1 a [K: the restriction of the permutation a to the set K ......................... 2 ST(Tr): the number of permutations in S, having exactly k occurrences of r ...... 6 g(T, a): the number of occurrences of 7 in a .................................. 9 g(7T, n): the maximum of g(Tr, a) over a E S, .................................... 9 p(T): the packing density of 7 ................................................ 10 gs(i, n): max{g(7r, a) :: a E S, is 1l,. i, I and has at most s lv. ~} .............. 19 A h 1: A is a partition of unity with s parts, ............ ....... ............ 20 p(T, A): the rcontent of the partition A of unity ............................... 20 q(w A): p A ) / () ........................................................ 21 ps(r): n ,, p(r, A) ................................. ................... 22 4K(r): the packing complexity of 7, ........................................ 23 e,..b (u, v): the sum E A A .................................. 24 a T,,,: the 1 .ii l pattern having li, r sizes (1",/) ............................... 34 a b: a divides b ............................................... 35 gk(1r,n): max{g (Tr, a) :: a E S, is of the form ALi...Lk and ILk > IA} ......... 44 g(H, a): the number of occurrences in a of members of the set H ................. 63 g(F, n): the maximum value of g(n, a) over all a S, ............................ 63 p(I): the packing density of the set I ......................................... 63 6(H, k, n): the maximum value of over all words a c [k], ................... 64 (m) 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 OPTIMIZING THE PACKING BEHAVIOR OF LAYERED PERMUTATION PATTERNS By Daniel E. Warren May 2005 C'!I ir: Mikl6s B6na Major Department: Mathematics Let r E S, and a E S, for m < n. We iv an msubset S {si,...,s,} c [n] is an occurrence of 7 in a if the subsequence usl, c2J,..., am is orderisomorphic to 7r, that is, as, > as, if and only if Ti > rj. We ~v 7 is l7.;, 1. .l if it can be decomposed into consecutive parts, called l r.zi~, so that its elements are in descending order within 1'. is and ascending between l r' is. It was conjectured by Wilf, and later proven by Galvin, that the maximum number of occurrences of a fixed pattern 7 is imptotically proportional to the number ('). This limiting ratio is called the packing 1. :,:/;l of T. A permutation a E S, is said to be roptimal if it contains the maximum number of occurrences of 7, over all patterns of size n. A theorem of Stromquist states that if 7 is 1 .i, 1. 1, we may assume that any roptimal permutation is also 1, i 1. i. We investigate the structure of the optimal permutations for containing occurrences of a few new families of patterns, and from this information we compute their packing densities. A permutation 7 is said to be of bounded type if the number of 1l'. ri in a r optimal permutation is bounded. We define the packing ,. i/pl ., ii, of a permutation of bounded type, a measure of the number of extra l rzi~ in an optimal permutation, and we compute bounds on it for a few classes of patterns. CHAPTER 1 BACKGROUND AND RELATED PROBLEMS This dissertation is a study of the behavior of occurrences of subpatterns in permutations, a subject which has a long and rich history and is related to several other problems within the realm of enumerative combinatorics. We begin with an overview of the necessary notation and terminology, after which we will present a brief survey of the research that led up to the inception of the permutation packing problem. 1.1 Permutations, Occurrences, and Layers In order to understand the results developed in the later parts of this document, we must first develop some basic vocabulary on the subject. In this document, the words permutation and pattern will both be used to describe orderings of the first few positive integers, viewed simply as words, e.g. 142365. The word p11 i ii" will generally be used to describe the smaller of the two. As per the usual notation, we will denote by [n] the set of the first n positive integers, and we will write S" for the set of all permutations of length n. In many cases it is easier to see what we are trying to prove if we have a graphic representation. For our purposes, the graph of the permutation 7 will be understood to be the set of points (i, r(i)) in the Cartesian plane. The graph of the permutation 34215876 is shown in Figure 11. Definition 1.1.1 Let 7r e S,. We /. I,:,' the reverse of r, denoted trev, by setting rev(i) = (n + 1 i) for i [n]. Definition 1.1.2 Let 7r E S,. D. I,.' the complement of T, denoted 7r, by setting Sr(i) = n + 1 i(i) for i [n]. 0 * ** ** * ___^___ Figure 11: The set {1, 2, 4, 7} is an occurrence of 2314 in 34215876. Definition 1.1.3 Let 7r e S,. The symmetry class of r is the set of all permuta tions in S, that can be found by 'r'l,';,:,', some sequence of the reverse, complement, and inverse operations to 7r. Notice that the symmetry class of a permutation may also be manifested graphically. The graph of any element in r's symmetry class can be achieved from the graph of 7r by a series of reflections across the lines x = y = and x y. In order for our study of the behavior of subpatterns to begin, we must first make clear what we mean by one pattern occurring inside another one. Definition 1.1.4 Let 7r E Sm and a E S, m < n. We .,;/ an msubset K of [n] is an occurrence of w in a if the restriction a K of a to the elements of K is isomorphic to 7, that is, the elements are in the same order. We i,,i'; also refer to the subpattern a FK as an occurrence of T. Example 1.1.5 As depicted in Figure 11, the set {1, 2, 4, 7} is an occurrence of 2314 in 34215876. A class of permutations which consistently reappears in the literature related to occurrences of patterns is the class of 1 , 1 permutations. Definition 1.1.6 A permutation 7 is called layered if it can be decomposed into an i ,,i, of descending sequences of consecutive integers, which are then ordered increasingly by first elements. Figure 12: 3215467 is the unique 1lit,. 1 permutation of type (3, 2, 1, 1). L ,rs are shown in shaded boxes. Alternately, we may describe 1 i,1, ,t permutations as those which have no descents of size greater than 1. Notice that a l i 1,. permutation is uniquely determined by its ordered list of l,,r lengths (its type); the 1 i ,. pattern of type (3, 2, 1, 1) is shown in Figure 12. That 1 ,i it, patterns are completely determined by their 1~.r lengths allows them to be counted easily. We may see now that the set of lit1, I patterns of size n is only a small subset of S,. Lemma 1.1.7 There are 2`1 ,,r;. ,,.. permutations of size n. Proof. Since each 1li, 1, ,l permutation in S, is determined by a composition of n, it suffices to count those. Write down n blanks in a row, choose a subset of the n 1 spaces between them in 2'1 v i and put bars in those spaces. The number of blanks between bars determines a unique ordered composition of n, which in turn determines a unique 11 l i permutation. The occurrences of I 1i, 1 patterns in 11i, It permutations are also easier to count, due to the following fact, the proof of which is left to the reader. Fact 1.1.8 Let m < n E Z+, and let w C Sm, and a E S,n be '7.;/, ,..l. Then, in every occurrence S of w in a, every '7.;/. of (a s must be contained in a 7.';/. of a. 0 0 0 0 0 0 0 1.2 Permutations and Posets The connections between permutations and partially ordered sets, or posets, has been exploited to some degree in the literature, so we will make a few definitions to make clear the relationships between the two that have been used. Definition 1.2.1 Let rT e S, be a pattern. D bo. the ypi'.,.:.ll'; ordered set P, to be the set [m], together with the partial ordering < /1, f,. 1 by i < j whenever i < j and 7r(i) < 7r(j). That is, i < j whenever (j, r(j)) is northeast of (i, r(i)) in the p'j'l1, of 7F. The poset P, for 7r 1327654 is shown in Figure 13. Definition 1.2.2 A post P is said to be layered if it can be partitioned into subsets P1,..., Pr, called 17.i,. so that i,.l: two elements of P are comparable if and only if they are in separate 7.';/. I,. Example 1.2.3 The post shown in Figure 13 is 7.'',. 1..l, with its 7.';, being {1},{2,3},{4,5,6,7}. The relationship between 1 1,i 1 permutations and 1 1,. 1 Il posets will now justify our use of a common terminology. Fact 1.2.4 A permutation 7r is 7.;/,. .1 if and only if the post P, is 7.';,. ,..l. We may define what it means for one poset to occur within another in a manner similar to our definition of an occurrence of one permutation inside another. Figure 13: The poset P, for 7r 1327654. Definition 1.2.5 An occurrence of a post Q inside a post P is a subset of P which is isomorphic to Q in the order inherited from P. Again, our new use of the terminology will be reflected in the given bijection between permutations and posets. Fact 1.2.6 Let 7 E S,, a E S,, n > m. The map a  Pi induces a bijection between occurrences of 7 in a and occurrences ~, in Pe,. The study of occurrences of 1 ,l. 1 Il posets within Il, 1. il posets will reappear when we arrive at the description of the permutation packing problem. To get there, we must start where the literature starts, with a problem in sorting. 1.3 Sorting Permutations with Stacks The problem from which the study of the occurrences of patterns in permu tations originally sprang, that of sorting permutations with stacks, arises in the listsorting algorithms encountered in elementary computer programming, but it can be easily represented by a problem in railyard design. Trains come into railyards, but the cars are in the wrong order; a stack, as shown in Figure 14, is one method by which some sorting can be done. Elements can be pushed onto the top of the stack or popped off the top. The question arose as to which permutations could be sorted by a given arrangement of stacks. There are two operations that can be performed by a stack: we can push an element onto the stack, or we can pop the top element off the stack. < OUT S IN STACK Figure 14: Swi' ird Stack Example 1.3.1 The permutation 54132 is sortable by one stack. To sort the sequence 54132, do push(5), push(), push(l), pop(l), push(3), push(2), pop(2), pop(3), pop(4), pop(5). Example 1.3.2 Every 7.';,. ,..l permutation can be sorted by one stack. Example 1.3.3 The sequence 231 cannot be sorted in one stack. As it turns out, 231 appears to be the only thing that causes the trouble, due to an early theorem of Knuth [12]. Theorem 1.3.4 (Knuth) A sequence of railroad cars can be sorted by a switch ,,,,/ of 1 stack if and only if the corresponding permutation does not contain wn1 occurrences of the pattern 231. Definition 1.3.5 If a has no occurrences of T, we .';/ a avoids T. Knuth's theorem thus states that the set of permutations which can be sorted by one stack is precisely the set of permutations which avoid 231. In fact, the problem of sortability in many cases has a solution in terms of avoidance, as exemplified by the following result of Tarjan [19]. Theorem 1.3.6 (Tarjan) If a sequence of railroad cars can be sorted by a switch :;/.,/ of k parallel stacks, it must avoid the pattern 2 3... (k + 2) 1. In general the converse of this theorem is not true. 1.4 Pattern Avoidance After Knuth's result that the permutations which are sortable by one stack are precisely those which avoid 231, a natural question that arose among the combina torics community was how to characterize, or at least count, these permutations. Thus began the enumeration of permutations by the number of occurrences of fixed subpatterns. There is a standard notation for the counting. Notation 1.4.1 Let 7r Sm, m < n. Denote by S,(r() the number of permutations containing ,.. I;. k occurrences of T. Explicit formulas for a few values of S,(Tr) are known. Theorem 1.4.2 (Knuth) Let 7~ be i,.:, fixed permutation of length 3. Then, Sow() (= the nth Catalan number. n+l' Theorem 1.4.3 (Gessel) We have 2i) (n 2 2 + 2i + 1 n 2in Si (i + )2(i + 2)(ni +1) Theorem 1.4.4 (B6na) We have 7n2 3n 2 (l (2i 4)! ( i + 2 v So(1342) 7232 (1) + 3 _: 2i+1.. (_1)" = 2 2 i!(i 2)! 2 Another main area of interest within the pattern avoidance question is the StanleyWilf conjecture. It was conjectured (independently) by Stanley and Wilf that the limit lim (S(7) existed for every pattern t. The sequence (So(Tr))n is called the StanleyWilf sequence of 7r. The StanleyWilf Conjecture is in fact a special case of a stronger statement about avoidance behavior in matrices whose only elements are Os and Is. Definition 1.4.5 Let A and P be matrices whose entries are either 0 or 1, and let P be of size k x We .',;/ A contains P if A has a k x submatrix Q so that if Pij 1, then Qij = 1 for all i,j. If A does not contain P, we .1;, A avoids P. Conjecture 1.4.6 (FurediHajnal Conjecture) Let P be i,;, permutation matrix, and let f(n, P) be the maximum number of 1 entries that a P'r; ... ;:,, matrix A can have. Then, there is a constant cp such that f(n, P) < cpn. It was proven by Martin Klazar [11] that the FiirediHajnal conjecture implies the StanleyWilf conjecture. His argument is also described in B6na's book [4]. The FiirediHajnal conjecture was proven in 2003 by Marcus and Tardos [13], so in particular the StanleyWilf conjecture is now known to be true. For a great while, in all cases where the limit of a StanleyWilf sequence was known, it was known to be an integer, so there was some speculation as to whether 8 this condition was alvi true. However, it was proven by B6na [5] that it is not in general the case that the limit is an integer; in fact, the limit of a StanleyWilf sequence need not even be rational, as shown by the following result. Theorem 1.4.7 ([5, Theorem 4.1]) Let k > 4, and let qk be the pattern 12 ... (k 3)(k 1) k (k 2) (so q4 = 1342, q5 = 12453, and so on). Then, we have lim (S(qk)) (k 4 + v8)2. nt CHAPTER 2 PERMUTATION PACKING In 1992, at a SIAM conference on Discrete Mathematics, Herb Wilf proposed an opposite question to that of pattern avoidance. That is, what if instead of attempting to avoid a fixed pattern 7r, we try to get as many occurrences of 7r as possible? How many can we get? The subject of permutation packing was born. 2.1 Definitions and Notation The largest body of work on the subject of permutation packing is the Ph.D. Thesis [14] of Alkes Price, whose notation we will adopt where we can. Notation 2.1.1 For Fr E Sm and a E S,, let g(Tr, ) be the number of occurrences of 7r in a. For each n E Z+, let g(x7, n) = max g(w, o). oES Definition 2.1.2 If o E S, and g(Tr,o) = g(T, n), we .;, a is Toptimal over S,. That g(T, n) < (') for every 7T c Sm is clear, since every occurrence of 7r in a is by definition an msubset of [n]. An early conjecture of Wilf was that g(T, n) was ..imptotically proportional to (n), and the following stronger result was later proven by Galvin. Lemma 2.1.3 (Galvin) The sequence is nonincreasing in n. Proof. We will show that )< 1) (rnI) (C 1) C('!..... a e S, to be Troptimal, so that g (r, n) = g(7T, a), and let S,_(o) be the set of subsequences of a of length n 1. Viewing each member of ST_(ou) as a permutation on n 1 elements, we may count the occurrences of 7r in a by Proof. Since g(,a) = g( ,a 1) = g(evrev) = g(, ), the conditions that a is roptimal, a is wloptimal, arev is wrevoptimal, and oa is Tioptimal are all equivalent. It follows that g (7, n) g (71, n) g(e rev,)= g (7C,) . The equality of packing densities follows. 4 The main objective of the permutation packing problem is to compute the packing densities of different classes of patterns. The most is known about patterns that are 1 lit ,1, so we will address those first. 2.2 The Impact of Layering An early result due to Stromquist gives us the distinct advantage that when 7 is 1 i ,. 1i, we are able to restrict our search for a roptimal permutation to the (much smaller) class of 1 l,. ri permutations on n. The result was proven in an unpublished typescript in 1993 [18] for the more general case of 1 li,. r posets, and the proof was reproduced by Price [14]. Theorem 2.2.1 (Stromquist) Let P be a fixed 7.';;, 1 Iposet. For i,; post Q, let g(P, Q) be the number of occurrences of P in Q. Then, we have max{g(P, Q) :: Q n} max{g(P, Q) :: IQ =n and Q is 1;,',;. ,,}, that is, among posets Q of size n which contain a maximal number of occurrences of P, there is one which is 7.',;. ,..l. Hence, the result in which we are interested, the one for 11Fi it permutations, follows immediately. Theorem 2.2.2 (Stromquist) Let 7 e Sm be /.;,. ,..l. Then, g(r, n) = max{g(7r, ao) :: Ca S,} = max{g(7, a) :: a E Sn is 7.';,. s .. }. The proof of Theorem 2.2.2 was also presented in a 2002 article [1] in the slightly more general context of packing a set of permutations. This generalization of the packing problem is addressed in C'!i pter 5. Theorem 2.2.2 aids us in two v . First, by Lemma 1.1.7, the number of 1 li. I1 permutations of size n is far less than n!, so our search space is smaller. The second advantage is gained from the result of Fact 1.1.8 that lz. i~ of subpatterns of a must be contained in l z. ?is of a. Fact 2.2.3 Suppose r has / .;, lengths (ml,... m,) and a has I ;,. lengths (ei,..., ~s) for some s > r. Then, we have _q(7,7)I M F2.1 1l i In general, the lack of this restriction is a large part of what makes the problem of finding packing densities of unrl li r Ip patterns so difficult. We will address the work done on ur i], ri I patterns in Chapter 5. 2.3 NearOptimality In many circumstances, it is difficult to prove that a particular family of permutations is Troptimal, but it is not too difficult to show that it is "close enough." To further describe this phenomenon, we introduce the notion of nearoptimality. Definition 2.3.1 Fix Fr 7 Sm. A sequence (o,) of permutations is said to be nearoptimal for containing occurrences of 7 if the limit lim g (, o.n) h\m is equal to the packing /. ,'.:/; of r. Notice that no single element oa, in a nearr optimal sequence need be roptimal over S,. Definition 2.3.2 A pattern 7r is said to be of bounded type if there is a nearTr optimal sequence (o,) such that the number of 7.;,. in oa, is bounded by a constant for all n. A result of Albert, Atkinson, Handley, Holton, and Stromquist allows us to find an easy way to ensure that we are working with patterns of bounded type. Theorem 2.3.3 ([1, Theorem 2.7]) A 1;,,. ,..l pattern 7r having no 7'.'; ,. of size 1 is of bounded ';I' A few special cases of patterns of bounded type warrant some special attention and have their own names. Definition 2.3.4 A pattern 7r is called simple if there is a nearTroptimal sequence (ao ) such that each permutation a, has the same number of 7.1,,. as Tr. Definition 2.3.5 A ,;~,. 1,, pattern 7r having k 17.;, , is called almost simple if there is a nearTroptimal sequence (a, ) such that each permutation a,, has ''. il; k + 1 1,';/. '. We close this chapter with a few examples of patterns for which the packing density is known, in order to motivate the more general techniques for computing it, which we will develop over the course of this document. 2.4 Expository Examples Example 2.4.1 We have p(2143) = . Sketch of Proof. That 3 is a lower bound for p(2143) can be obtained from looking at the permutations ,2 2(n) = nn ( 1)... (1) 2n (2n 1)... (n + 1) ** Figure 21: Longer permutations having two lrz,i of equal length correspond to the partition of unity having two equal parts. 0 Fiue :Logr emtainshvngto .rso eullegh orspn t0h patiio o uit hvig woeqalpa0s of size 2n. Certainly g(2143, 2,2(n)) (,)2, as each 1 vr of 2143 must be contained in a l1 ir of a,. In this case we get the lower bound S g(2143, a,) p(r) > hlim 2n n \4 S(n) 2 = li n (2n) 44 / n2(n1)2 lim 22 Sm 2n(2n1)(2n2)(2n3) 4! n4 lim 4 24n4 24 6 3 24 8 It can now be proven that this lower bound is tight; this task can be accomplished by proving that the permutation a2,2(n) is in fact 2143optimal in S2,,. First, we show that there is a 2143optimal permutation in S2. having only two l1,ri, and then it may be shown that among twoi i. 1 permutations, the 2143optimal one has two lI..ir of the same length. We may begin to describe an intuitive explanation of another technique that is often applied in looking at patterns that behave similarly to 2143. Since we are looking at larger and larger permutations, consider what happens when we scale each one to fit inside the unit interval [0, 1]. Longer and longer permutations having only two lrzis begin to correspond to partitions of the unit interval into two parts, as shown in Figure 21. Suppose we can show that there is alv,v a 2143optimal permutation of size n which has only two lIvr C'! .... a ,, E [0, 1] so that the lIi.r lengths in such a permutation are a, n and 0 n. It turns out that the sequences (a,) and (n) converge, and we have (n2,n) (/l3,n2 p(2lt3) lim 2 2 C (n4) (2) (0a2Q where a = lim, a, and p = lim, /,. We may therefore calculate p(2143) by maximizing the polynomial ()c a2/2 over the set of pairs (c, 3) such that a, 3 E [0, 1] and a + 3 1. Let p(2143, A) denote the polynomial ( ) A2A The partition A = (c, 3) is called a 2143optimal partition of unity, and a and 3 are called the asymptotic layer ratios for a 2143optimal permutation. In the present case, the maximum occurs when a = 3 so that the packing density of 2143 is calculated to be (4) (1)2 (1)2 This technique of determining optimal .i,iil H1 c ._ lrC ratios by maximizing a polynomial over the set of partitions of unity was pioneered by Price [14] and further explored by Hasto [8]. We will learn more about this technique in C'! plter 3. Example 2.4.2 We have p(132) 2v 3. Sketch of Proof. Let a be a 132optimal permutation of size n. By Theorem 2.2.2, we can assume a is 1,i,. 1, Let L be the last ly r of a, and write a = a*L. An occurrence of 132 in a must either consist of two elements of L and one element of a* or be contained entirely in a*. We have no stipulations on what a* is, so we may as well assume that it is a 132optimal permutation of smaller size. We get the recurrence g(132, n) max g(132, k) + k n k 2 k It may then be proven by induction that any number f(n) that satisfies the recur rence (2.2) must also satisfy n3 3 (2/3 3) 52 < f(n) < (2/ 3) , 6 6 so that the packing density is computed to be 23 3 by ordinary means. Though the above proof is sufficient to prove the value of p(132) once it is guessed, it fails to provide any information on why we chose the number 2V3 3 in the first place. To this end, we will try to apply the same technique here that we previously applied to the pattern 2143. Suppose A is a 132optimal partition of unity. Let Ao be the farthest part on the right. Then, the recurrence (2.2) i that the remaining partition of (1 Ao) must also be 132optimal. We may as well assume, then, that it is a scaleddown "li, of the entire partition A, which would correspond to a permutation structured like the one shown in Figure 22. This assumption leads to A having an infinite number of parts which decrease geometically in size from the right. Number the parts of A with the elements of N, starting from the right. If Ao has size a, then Ai must have size a(1 a) for all i E Z+. An occurrence of 132 in a permutation with 1v.r ratios A1 ... Ak corresponds to a triple (x, y, z) of points such that x E Aj and y, z c Ai for some i < j. It follows that the packing density of 132 should be the maximum of the polynomial p(132, A) = (3) Eoi Then, we have p(132,A) 2) AAJ 0 i=0 j=i+l 3 j +2 1 ) 3(1 )3 2i i=0 I=i+1 DC F i+1 i3( (O ) io 3(1 _F)2Y(F 3)i tO S3E(1 )2 ( 1 3) = 1 )2 Figure 22: Structure of a 132optimal permutation. The boxed area is a scaleddown < i, of the whole permutation. The packing density of 132 should therefore be found as the maximum value of the function () 3x(1 x)2 h(x) = 3 1 X3 for x e [0, 1]. Setting the numerator of h'(x) to zero, we find that the value of x in [0, 1] that maximizes the value of h(x) is xo = , so we can compute the packing density p(132) = h(xo) = 23 3. There are two important lessons to take from this example. First, the packing density of a pattern need not be rational; in this case it is not. Secondly, when 7 is 1 ,li. ,. 1, while we may assume that each roptimal npermutation a, is 1 ., li1, there is no guarantee that the number of lV. ir in a, is bounded by any finite number. A pattern 7 is said to be of unbounded type if every nearoptimal sequence for 7 contains permutations having arbitrarily many 1 vir. Such permutations are addressed in C'!h pter 4. Example 2.4.3 We have p(1243) = . Sketch of Proof. That 3 is a lower bound for p(1243) can easily be proven by looking at the permutations a, = 1 2... n 2n (2n 1)... (n + 1), as g(1243, 2n) is then (2) , as in Example 2.4.1. Notice, however, that while the 2143optimal permutation in Example 2.4.1 .l1 i, had two l1VIis, the permutation a, has n + 1 t1V is, so 1243 18 is of unbounded type. It was proven in 2002 [1] that the permutation oa, above is actually 1243optimal for each n, but the proof was not trivial. We will generalize this result to a family of similarlystructured patterns in ('!i lpter 4. CHAPTER 3 PATTERNS OF BOUNDED TYPE In this chapter, we will discuss the packing behavior of patterns having optimal or nearoptimal permutations with a bounded number of Ilv. r. The behavior of these patterns will be generalizations of Example 2.4.1. To better understand the way with which these patterns are dealt, we first describe a framework in which the 1 1, i, patterns can be viewed, conceived by Alkes Price, in which we use the language of partitions to look at large permutations with a bounded number of 3.1 Permutations, Partitions, and Complexity Notation 3.1.1 Let 7r E Sm be a 7..';,. '.. pattern with r 7.;,. , of sizes mi,... ,mr, and let s > r E N. For n > m, let gs(i, n) = max{g(7, o) :: a E S, is 7.';,. ,..l and has at most s ~7.';. '}. The following Lemma follows from Fact 2.2.3. Lemma 3.1.2 (Price) For r, n, and s as above, we have gs(7, n)= max 1i,...eN m \ Mr l++e,=n Zli << Fact 3.1.3 (Price) The sequence (gs(7,rn}) is increasing and ,' ', 'la con stant. Proof. That gs(7r, n) < gs+l((r, n) for all s is clear, since any permutation with at most s 1i ri also has at most s + 1 Il1. ri. Since a permutation of size n can have at most n 1 ?ivi, for s > n we have gs(, n) = max{g(7, o) :: a E Sn is 11,, r, }, which is equal to g(7, n) by Stromquist's Theorem (2.2.2). To characterize the .ivmptotic behavior of the numbers gs8(, n), Price used a probablistic technique. The idea is to simulate the behavior of large permutations with a fixed number of lI1i, using partitions of unity which describe the ratios of the livr lengths. Intuitively, the pattern a2,2(n) defined in Example 2.4.1 should be accurately described by the partition ( ) since it alvi, has two lir. rs which are half the length of the permutation (see Figure 21). Notation 3.1.4 We write A , 1 if A= (A1,... A,) e [0, 1]" and A1 + + A, 1. Let A h 1, and let 7 be the 1li 1,. permutation of type (ml,... ,mr). Let m = mi + + m,. Suppose we have a family (an) of permutations such that (a) On c Sn, (b) every a,, has s l iSr~, and (c) for i = 1,..., s, the size of the ith lvr of a,n is LAin] (the closest integer to Ain). Then, for each n, we have (m 1 i<" so taking the limit in n we have AmI ..Amrnm mi,. ,A . lira 9(7,) K m li S Am!. ...r! S I""" 1 Definition 3.1.5 We /7;,,: the rcontent of A to be p(7T,\):( n Y A ... Ar. T'I ."",Tnr 1< < .<... <{ < 8 The partition function p(T, A) is thus used to describe the limit packing behavior of 7 into permutations whose sequences of li.r lengths have a similar structure to A. 0 1 I I I I I I I A1 A2 A3 A4 A5 A6 Figure 31: Here, (2, 3, 4) and A 6 1. The function q(r, A) gives the probability that the first two balls occur in a box Bi, the next three balls occur in a box Bj, and the last four balls occur in a box Bk, for i < j < k. Notation 3.1.6 In some cases, it will be more convenient to work without the multinomial coefficient, so we will write q(7, A) for p(, A)/(m1,,) ). The function q(7, A) also has a useful probablistic interpretation on its own: Suppose we have m distinguishable balls in r colors, where the first m are color 1, the second m2 are color 2, ..., the last m, are color r. Divide the unit interval up into s boxes B1,..., B, such that box Bi has length Ai. Now, throw the balls randomly at [0,1]. For each set ii,..., i, C [s], the probability that the first mi balls land in Bi,, the second m2 land in Bi,, ..., and the last m, fall in Bi, is exactly A1" A"2 ... A Hence q(7, A) is precisely the probability of the following event, depicted in Figure 31: 1. All balls of the same color are in the same box. 2. All boxes are monochrome or empty. 3. The numbering of the colors increases to the right, that is, all balls of color i are found in boxes to the left of all balls of color i + 1 for each i. As we are interested in limiting the number of l, ri in permutations containing 7, in this context we may wish to know the maximum value of the partition function p(7, A) when we restrict the number of parts in A. Definition 3.1.7 Let ps(r) maxp (7, A). A 1I The following theorem of Price now provides the necessary relationship between the partition functions and the numbers gs(r, n). Theorem 3.1.8 ([14, Theorem 3.1]) For every s, the limit lim, Y(') exists and is equal to ps(r). Since g(Tr, n) > g (Q, n) for every n and s, it now follows from taking limits that p(t) > ps(7), in particular p (r) > p (T, A) for every A. However, the next result of Price tells us that the approximation of p(7) by partition functions can be as close as we want. Theorem 3.1.9 ([14, Theorem 3.2]) Let 7 be the 7.';,. ,,l pattern (m1,... ,m). Then, we have limps(7) = p(). It is important at this point to note that the set of partitions of unity having s parts (some of which may be zero) is a compact subset of [0, 1], so we are justified in writing max instead of sup in Definition 3.1.7. That is, for each s, there is a partition As such that ps(7) = p(r, As). In the case that 7 is of bounded type, there is a finite bound to the number of lv. ri occurring in the elements of a nearToptimal sequence, so the sequence (ps,()), is constant after a finite number of terms. It follows that there is a number k and partition A K+k 1 such that p(7) p(7w,A). 3.1 In a natural extension of our terminology, such a partition is called roptimal. This connection between nearoptimal sequences and optimal partitions was explored in 2002 by Peter Hasto. Lemma 3.1.10 ([8, Lemma 2.1]) The permutation 7 having r 7.';,. , is simple if and only if there is a roptimal partition A with .. i11,; r parts. 7 is almost simple if and only if there is a roptimal partition A having '.'. i/l; r + 1 parts. We extend this notion to define a measure of how far from simplicity a 1 I, 1 permutation is. Definition 3.1.11 Let 7r be an rl,;,. pattern of bounded i';i. We f;,': the packing complexity of r, denoted K(7r), to be the minimum k such that there is a 7roptimal partition A ,+k 1 By Lemma 3.1.10, simple permutations thus have complexity zero, and almost simple patterns have complexity 1. Later in this chapter we will find bounds on K for some other classes of patterns. 3.2 Simple and AlmostSimple Patterns Particular attention has been paid to the case when there is a nearroptimal sequence of permutations having the same number of liv i as 7r. The packing behavior of these patterns was totally characterized by Price. Theorem 3.2.1 (Price) Let r be the '7.;,. ,'. pattern (mi,... m) E S,. If T is simple, then, the partition A = (i,..., m) is 7optimal, so the packing /. ,.'/; of4 is p (7 ) = p (7 \A ) m m  S/ m,...,mr \m/ \m/ Since the packing density of any simple pattern is known, we may then wish to know which patterns are simple. Price proved that every 1 li,. 1 l pattern consisting of two livri of size at least two is simple. Theorem 3.2.2 ([14, Theorem 5.3]) Let r be the '7.;,. ,.l pattern (a,b), where 2 < a < b. For each n, there is a 7roptimal permutation a E S,, which has only two . ;1', Hence, 7~ is simple, and S(() a + b a b b In 2002, Peter Hast6 [8] was able to generalize this work and describe a sufficient condition for simplicity. Theorem 3.2.3 (Hist6) Let 7r be the '7.;,. ,..l pattern (mi,... m) in S,. If log2(r + 1) < min{mi}, then ~ is simple. To complement this result, Hast6 also proved a sufficient condition for nonsim plicity of supersymmetric patterns. Lemma 3.2.4 ([8, Lemma 3.5]) Let r be the 7.';,. .I pattern (p'). If log(r + 1) rlog(1 + ) then r is not simple. Another result of Hastb [8] is an example of a specific class of almost simple patterns which each contain a singleton l1.r. In particular, this result shows that the sufficient condition given in Theorem 2.3.3 is not necessary. Theorem 3.2.5 (HiistS) For k > 3, the 7.';,. l pattern 7k = (k, 1, k) is almost simple, and we have /2k +1 k2k + (k/2)k (2k + 1 k2k + 2kk k, l, k (2k + l)2k+l p k, 1,k (2k +l)2k+1 3.3 Increasing Patterns In this section, we will find an upper bound on the complexity of an increasing 1lit v I, pattern, that is, a 1,, li Il pattern r of type (mi,. r) such that 2 < m1i < .. < mr. In the case that these permutations are not simple, the summation notation encountered in Definition 3.1.5 becomes cumbersome, so we will borrow some of Hasti's notation [8] to simplify our computation. Notation 3.3.1 Denote S(uv) AX"U... A7. a..b a Fact 3.3.2 We have 1..s In fact, there is a probablistic interpretation of the formula Da..b (u, v) in the same context as the function q(r, A) when we only look at a subset of the colors. 0 1 I I I I I I I A1 A2 A3 A4 A5 A6 Figure 32: Here again, 7 (2, 3, 4) and A h 1. The formula 3..5 (2, 3) gives the probability that the third, fourth, and fifth balls occur in a box Bj and the last four balls occur in a box Bk for some 3 < j < k < 5. The set of balls and boxes to which we are restricting our attention is shown boxed. Suppose we only look at the balls of colors u through v, that is, the first m, of our balls have color u, ..., the last m, balls are of color v. Throw them at the unit interval as before. Now, ea..b (u, v) computes the probability that the same three conditions hold as before, with the added condition that all the balls we throw (just those of colors u through v) land in the boxes B,,..., Bb, as shown in Figure 32. This interpretation of ea..b (u, v) leads to the following useful formula: Fact 3.3.3 We have e( r) ( (tr)+Ar @ (,r ). 1..s 1..s1 1..s1 Proof. When throwing the n balls at the unit interval, either the last m, balls go in the last box (and the rest go in boxes B,... B81), or all the balls go in the first s1 boxes. In order to simplify the proof of the main result of this section, we will separate out a couple of technical facts we will need and prove them as lemmas. The following inequality will pl iv a key role in our inductive proof of a result for increasing 1 li, t , patterns. Lemma 3.3.4 Let 2 < mi < ... < m,, and let A 1 such that Ai < ... < As. Then, we have At2 @ (3, r)_< A (@ (2, r ). 3.2 3..s1 3..s1 Proof. Notice that the sums on both sides have the same number of terms, and in fact the terms of both sums are indexed by the same set, namely subsets of {3,...,s 1} of size r 2. For a set A = {al,...,a,2} C {3,...,s 1}, there is exactly one term on each side of (3.2), i (A) on the left and r(A) on the right, which contains all the Aa, for i = 1,..., r 2, raised to some powers. On the left, we have (A) = A2 A3... A2, and on the right, we have r(A) A2 ...Am' Am Now, we know that for each j = 1,...,r 3, we have 2 < aj < aj+l < s so A2 < Aaj < Aai s< AS since A is increasing. Therefore, we have Aa Aa2 ,Aa,2 A, A (X^2 ( 3 (r3< r It follows that f(A) < r(A) for each set A C {3,..., s1}. We then get the inequality (3.2) term by term. Hast6 [8] proved the inequality A2 (3, r)_< (r ) @ (2, r), 3 3..s 3..s but ii.:: 1. 1 that the inequality (3.4) below is true. We can now use Lemma 3.3.4 to prove (3.4), using a very similar proof to that of (3.3). Lemma 3.3.5 Let 2 < mi < .. < m,, let s > r, and let A s 1 be an increasing partition as above. Then, r e(3, r) Proof. The proof is by induction. If s = r + 1, then s r = 1, so the statement is exactly that of Hist6. If r = 2, we must show that 1 A72 < (AT2+...A2). s2 However, since A2 < Aj for j > 3, it follows that A"2 is no bigger than any term on the right, so it can be no larger than their average. Assume now that (3.4) holds for s 1, for r = 2,..., s 2, and that r < s 2. By induction, the case (s 1, r) gives us (s 1 )A2 A (3,r) < (r ) (2,r) 3..s1 3..s1 and the case (s 1, r 1) gives us (s (r ))A72 ( (3, r ) < (r 2) (2, r 1). 3..s1 3..s1 Since e (3,r)= (3, r)+ A7 (3,r ), 3..s 3..s1 3..s1 it follows now from Lemma 3.3.4 that (s ) )A72 (])(3,r) (3,r) + (.s + r)AA7 A (3,r 1) 3..s 3..s1 3..s1 ((s r )A2 @ (3, r)+ A2 @ (3,r) 3..s1 3..s1 +(s r)A 7 A7@ (3, r 1) 3..s1 (r 1) (2,+r) A'@ (2,r 1) 3..s1 3..s1 + A"'(r 2) (2,r 1) 3..s1 = (r 1) (2, r) + A (2, r ) 3..s1 3..s1 (r ) @(2,r) 3..s The inequality (3.4) follows. While we know from Theorem 2.3.3 that increasing patterns with no singleton 1lwri~ are of bounded type, we will need an extra result [8] which further charac terizes the structure of a 7roptimal partition A, namely that A can be taken to be increasing as well. Lemma 3.3.6 ([8, Lemma 3.2]) Let 7r be a 1',;/, .. pattern of type (mn,... ,mr.) such that miI < mn2 < ... A 1 such that Ai < A2 < ... < A, and p(7r) p(r, A). Now, we are ready to prove the main result of this section, a generalization of Theorem 3.2.3 to an upper bound on (Tr). Theorem 3.3.7 Suppose 7r is an increasing I7.;,. .l pattern of type (ml,..., m,) with no singleton 7.';/., that is, we have 2 < mi < .. < mr. If mi > log2 (rk + 2), then K(7) < k. Proof. The proof here is just a small generalization of the proof [8] of Theorem 3.2.3. By Lemma 3.3.6, 7r is of bounded type, and there is an increasing optimal partition A s 1 (choose A such that s is minimal). Suppose K(T) = s r > k. Since s is chosen minimally, there is no 7roptimal partition of size s 1, so we have q(7, (A1 + A2, A3, As)) < qs_1w7) < qsw) = q(7, (A1, A)), which after some rearrangement of terms translates to ((A + A2)" AT1 A1) (2, r) < A1A2 e (3, r). 3.5 3..s 3..s By (3.4), the inequality (3.5) becomes r1 ((A1 + A2)" Ar A) (2, r) < Ar (2, r). s r 3..s 3..s The sum is nonzero, so we may divide both sides of the above expression by it and A'1, resulting in the inequality ( AI2 ( 2 s Since the lefthand side has its minimum at A1 = A2, we have r1 r1 2'1 2< < sr k But then ml < log2 (k + 2) contradicting the hypotheses. Notice that substituting K = 1 reproduces Hast6's result. The result is perhaps better presented as a direct upper bound on K: Corollary 3.3.8 Let r = (ml,... m) such that 2 < mi < ... < m,. Then, K(7) <  2 3.4 Supersymmetric Patterns A supersymmetric pattern is al li1, I pattern whose lIri all have the same size, that is, a li. r, i I pattern of type (pr). As permutations having .,,li i,:ent singleton lIri are suspected to never be of bounded type, we will assume here that p is at least two. That (pr) is of bounded type when p > 2 follows immediately from Lemma 2.3.3. In this section we will provide a lower bound on the complexity of a supersym metric pattern that will generalize Lemma 3.2.4. To better characterize the nature of these patterns, we will need a little more power, namely the following two results due to Price: Lemma 3.4.1 ([14, Lemma 6.2]) Let 7r be the 7.';. ..1 pattern of 1;/'.' (p') for some p > 2. For ii,.1 s > r, if A F 1 is the partition i,,. i,,.:i .:,i.I p(7r, A) over all A F 1, then i,;ii two nonzero elements of A must be equal. Lemma 3.4.2 ([14, Theorem 6.1]) Let p > 2, and let h(s) =(8) ()r ". Then, (h(s)) is unimodal in s and ev, l.i,,lli decreasing. Lemma 3.4.1 narrows our search to the realm of supersymmetric partititions A, which are much easier to understand than the general case. In particular, if r = (pr) and A = () then we have GI P /1 S ( t )vrp From here, Lemma 3.4.2 gives an explicit characterization of the behavior of p(7, A) in the case that A and 7 are both supersymmetric. We can now prove the main result of this section. Proposition 3.4.3 Let w be the 7.;,. 1. pattern of ';I'. (pr) for some p > 2. If log2 ( + ) 3.6 Slog2 ( + k r then K(r) > k. Proof. Suppose (3.6) holds. Then, clearing denominators and logarithms, we have S+ < 1 + r ( k+r) ( k+1 that is, (r+k+1 r+k+1 (r++1) r+k k+1 (rk) It now follows that (r+k) 1 t (r+k+1 1 ( : r r+k r r+k+) ' so the sequence (h(s))7 is increasing at s r + k. Since this sequence is unimodal and eventually decreasing by Lemma 3.4.2, it follows that it cannot have a maximum at or before s = r + k. Now, since 7 is of bounded type, Lemma 3.4.1 together with (3.1) provide that p(T) is ( r7" ) times a member of the sequence h(s), naturally an element of maximum value. We have shown that the value of s for which this maximum is achieved is bigger than r + k, that is, K(T) > k. Notice that substituting k = 0 yields exactly Hast6's result. In order to gauge the accuracy of the bound in Corollary 3.3.8, we would like to show that there are permutations whose complexity comes close to that bound. To that end, we have the following result. Corollary 3.4.4 For p > 2 and r > 6, we have K(pr) > 3 1. Proof. Let k = 3 1. Then, we have +1 3/'1 log2 (1 + 3" 1) = log2(3). Furthermore, for every r we have 1 1+ k+r 1+r 3Gr 3/ + 1t) 3r 3, rlr For r > 6, we have (r )r < 3, so that in particular r log2 + k < log2(3). Hence for r > 6 we have log2 ( + ) 10log2(3) p r1 log g2(3) so ,W) > 1 3( 1 by Theorem 3.4.3. Note also that since limroo (r)r = e, the only property of the number 3 that we used for this proof was that it was larger than e, so any number of the form e + (E > 0) may be substituted in the above proof as long as we exchange the number 6 with a suitably large number in the statement of the result. We also have proof now that there are supersymmetric patterns with arbitrarily high complexity. Corollary 3.4.5 Let p > 2 and let n E Z. For suff,. ,l, li,,la1. r, we have ,<,i > n. Proof. C'! i..... r > max{6, (n + 1)(31 1)}, we have > r 1 > n. 3/ t Example 3.4.6 There is an r so that K(11r) > 100. Proof. C'!..... r > 101(311 1) 17891746. 3.5 A Partial Order on the Layered Patterns in S, A general trend in the literature is that it appears to be the lIis which are smaller that cause permutations to have greater complexity, whereas those with less numerous, larger 1,. ir tend to occur more toward the simple end of the spectrum. We may notice here that it is easy to extend the definition of K to all 1 ,li 1, permutations by defining the complexity of a permutation of unbounded type to be +oc. There is a natural partial ordering on the set of 1li ,. permutations in Sm, which we may take from the refinement order on ordered compositions of n. That is, we let w< 7 if every 1i.r of is contained in a li r of T. For example, 1243 < 2143, since the lvr decompositions are 12143 and 2143. Certainly (((1")) = +oo, and (((m)) = 0, and we have (1") <4 < (m) for all 1,i, 1,l e S,. Examples indicate that in all known cases the ordering of K is an extension of the dual of the order <. These observations have led me to make the following conjecture: Conjecture 3.5.1 Let ( and 7 be 7.,';. ,. .I permutations. If ( < r according to the order just described, then K(0) > K(T). CHAPTER 4 PATTERNS OF UNBOUNDED TYPE In this chapter we will explore the packing behavior of patterns which are not so wellbehaved as those in the previous chapter, that is, those whose optimal permutations have an everincreasing number of lIvi. All results in this chapter concern patterns of a common form, those which can be decomposed into two blocks, one increasing and one decreasing. We begin by standardizing the notation. Notation 4.0.2 Denote by 7T,>, the 7.',;. ,. pattern having 7.';, , (1",/ ). 4.1 The Layered Pattern 7,,3 In the case that a = 1, the pattern 7,,,3 is just the pattern 1 (/ + 1) 3... 2. Such patterns were proven by Price to exhibit similar packing behavior to that of the permutation 132 in Example 2.4.2. Theorem 4.1.1 (Price) Let 3 > 2. Then, P ( )/, (1 _)31 where E is the unique root of the p ...1;,,...;;:,,: q(x) = Ox+1 ( + 1)x + 1 in [0, 1]. In the proof, it is shown that there is a T ,,optimal permutation of size n with a fractal design similar to that encountered in Example 2.4.2. Theorem 4.1.1 was reproven in 2003 [9] by proving direct bounds on g (r ,n). Theorem 4.1.2 ([9]) Let = fe(1 E)~1, where E is the unique root of the j,..1 ;,,.,,,;l q(x) = Ox+1 ( + 1)x + 1 in [0,1]. Then, for all n > > 2, we have (n )+1 (n + 62,hoY+1 (f+ 1)! g ( ) +< (+ 1)! where 62, is the Kronecker delta. 4.2 Layers and Antilayers For a > 1, we no longer have fractally structured optimal permutations, but we run into the trouble caused by consecutive 1 ri~ of size 1. It has long been thought that the optimal packing behavior of patterns containing a contiguous block of r li.i, of size 1 was analagous to that of patterns having one livr of size r in the corresponding place. The pattern 22 = 1243 of Example 2.4.3 was the first pattern having this property for which the packing density was computed [1]. It was shown that the permutations optimal for 1243 are analogous in structure to those for 2143, that is, they consist of a single increasing block followed by a single decreasing block of the same size. In the remaining part of this chapter, we provide a generalization of their work to the patterns T,, such that /3 > a > 2 and a 2/3. To deal efficiently with contiguous blocks of l,_is of size 1, we make a few definitions. Definition 4.2.1 DA fi,' an antilayer to be a i,, ni.:ni,,lli sized sequence of consecu tive 7.';/. , of size 1, that is, an increasing sequence of consecutive terms. Example 4.2.2 The third, fourth, and fifth elements of the pattern 21345876 form an i,,.:li;;. of size 3. Fact 4.2.3 When translated into the lau,o.i,rg.: of posets via DA ili..i 1.2.1, l,.r; and ,,IJ.:l ;,N. in 7 correspond to antichains and chains in P,, '. p".. /.:;. 1;, Example 4.2.4 The pattern r,,, consists of an ,..:l,.r, of size a followed by a ,7I;/. of size 3. Definition 4.2.5 The more general term block will be understood to mean either a 1.;, or an i,;../.:l~, ,. Hence 2block patterns are those which either consist of two 17.;. i or a one '7.'/. and one ,'../.:;, t, . Definition 4.2.6 D fio, isolated point in a pattern 7 to be a singleton 7. ;, between two '7.,;. ', of ir ,' sizes. Example 4.2.7 The fourth element of the pattern 3214765 is an isolated point. Figure 41: (a) An occurrence of the pattern 7 = 32154 in a 5 4 3 2 1 7 6 10 9 8 is circled. Notice that every lv,cr of 7 is contained in a 1 . r of a. (b) An occurrence of 7r 12543 in a 2 1 4 3 6 5 10 9 8 7 is circled. See how the antilayer of 7 may "climb" several li, r of a. While the concepts of a l,r and an antilivr look virtually the same, they unfortunately must be handled in quite different viv, given the traditional approach to the problem. Due to the work of Price and Stromquist, when r is a 1 ,li 1, 1 pattern, we are able to restrict our search for rmaximal permutations of [n] to the much smaller class of 1 li 1. permutations of [n]. Note, however, that in an occurrence of the pattern r in a permutation o, any 1vr of r must be contained in a 1cr of a, but antili.,ir in need not be contained in anti .i. rs of o, as they can "climb" a list of several l.,i, having one element in each (see Figure 41). This simple problem is enough to make the computation notably more difficult, so that a slightly more delicate argument is required to prove a conjecture which at the outset seems obvious. 4.3 The Layered Pattern r; , The idea in this section, which will reappear later in this chapter, is that given the packing behavior of a small pattern, we may determine the packing densities for larger, similar patterns by exploiting their similarity in structure. Proposition 4.3.1 The structure of the in,, .,,.:..:,., permutation of size 2n for the pattern 7;r, is invariant of a, that is, the 7r ,,r ,,..:. ':,j. pattern of length 2n is the pattern r . Proof. The case a = 2 was proven by Albert et.al [1, Proposition 2.4]; it follows that the maximal number of 1243s in a pattern of length 2n is (1)2. In each step, we will show that if the pattern r,,, is not T7o, maximal, then in fact it cannot be 72,2)maximal, which would contradict the known result for 1243. We will first prove the a = 3 case. Consider 733, = 123654, and suppose that g 733,, 2n) > () 2 + 1. Let a E S2,2 be 7,3) maximal. In each 3,3), there are 9 ()2 instances of 1243s. Suppose that o is an instance of 1243 in a. Now, each of the remaining 2n 4 elements of a, if in a 7,3) containing To, can be on either side, the 123, or the 654, but not both1 , so there are f (2n 4 ) v,v to form a 73,3) for some Since the largest value of the expression f (2n 4 ) occurs when = n 2, it follows that the number of occurrences of 73,3) containing a particular occurrence of 1243 is at most (n 2)2. Hence, if a has at least (,)2 + 1 occurrences of 73,3), then it must have at least (3)2 \2 2 Sn +1 > (n 2)2 [3 > occurrences of 1243, contradicting the known result [1]. The general case is quite similar. Suppose that g r,, 2n) > (> ) + 1 and let a E S2n, be a permutation having g (T ,, 2n) occurrences of T ,,. We will again count the number of 1243s in a. In each occurrence of T,,>, there are (a)2 occurrences of 1243. Given a particular occurrence To of 1243, we need to determine the maximum number of 7; s which could contain To; however, for each T, containing To, the other 2a 4 of its elements must be an occurrence of 7; 2,, 2 in the remaining 2n 4 elements of a. That is, the number of occurrences of 7,,> which can contain T is bounded above by g ( 2,2 2, 2n 4). By induction, there are at 1 This fact follows from the fact that a can be assumed to be 1 , iI, 1. See the proof of Theorem 4.7.2 for a more detailed treatment of the general case. most (' )2 of these occurrences. Hence, the number of 1243s in a is at least a2 (n2) 2 a2+ a2 ) ) where > 0, which again contradicts the base case [1]. It follows that g (7,T 2n) < (T)2 for each n, which means that ,,,, is in fact a R7,)> maximizing permutation of length 2n. Corollary 4.3.2 The packing /. ,,:u'; of the pattern r,,> above is 22a Proof. The result follows from taking the limit lim lim (a g 2n) 2n n2. lim (a!)2 = hm^ n 22Tn2. (2a)! (2a)! (a!)2 22a In the next section, we will prove a statement about patterns of the same form, but with two blocks being permitted to have different sizes. 4.4 The Layered Pattern r72,p The reason that the packing behavior of 7,,> was so easy to characterize was its heavy degree of symmetry, namely that the lIvr and the antiliv r involved were of equal size. As the optimal packing behavior of the 1 iv. 1 l pattern T,, does not in general adhere to the same degree of symmetry as the case a = the ideas of the previous section do not so easily lend themselves to the general case. Hence, for the duration of this section we must cease to rely on known results and do some computation by brute force. We will later need the following technical lemma. Lemma 4.4.1 Let k, e,m,n E N be such that k < < m < n. Then, we have k f f k That is, we can have more combinations if bigger sets choose the bigger subsets. Proof. Straight computation: we have ( ) k! ) (m ) (... )(Tm 1) (k) and (k) k! (n 1( )() f + 1), so the result follows from crossmultiplying. For the remainder of this section, let3 > 3, and let a E S, be a permutation which is T2, maximal. It follows from Stromquist's theorem that we may assume a is 1 li 1, r 1. The in j o ii ly of this section will be justification for a series of assumptions we can make about a, until we can reduce a to an easy form. Lemma 4.4.2 We can assume without loss of genetii./li that a has no isolated points. Proof. Suppose we have an isolated point in between 1?, i Li and Li+l. Switching the positions of Li and the isolated point does not decrease the number of occur rences of T, so we may as well do that. Since we can ahivx move an isolated point to the left of a 1 .v r, eventually all isolated points will be soaked up by antiliv . We can assume, then, that a has no isolated points. It follows now that a is a sequence of 1lV ir and antiliv, rs, with no isolated points. In fact, a more general statement is true by the same logic. Lemma 4.4.3 We i,, r; assume that a consists of a single inJ.:l; r. A followed by a list of nontriviall) .;. L, Lk, as shown in Figure j2. Figure 42: The form of a as a single antili v.r followed by a list of l1irT. Proof. Suppose that in the middle of the permutation a we have an .ilil ii er between two l. ir. Similar to the proof of Lemma 4.4.2, if we switch the positions of the antil .r and the l.r to its left (see Figure 43), all the occurrences of 2,,>, that were originally there are left intact, and we create more as long as the antii r has length at least 2 and the left livr has length at least 3. Thus, in a permutation , we may move all .ili ii ,vers to the left of all r1vsi without losing any occurrences of 72,3,, achieving the desired 1'vir/antil ivr pattern. Notation 4.4.4 For a permutation in the format of Lemma 4.4.3, let a be the size of the ii .:l~;, A, and for each i, let fi = ILi. Lemma 4.4.5 The ~r; , L1,..., Lk i,,m'i be assumed to be in nondecreasing order by size. b Figure 43: There can be no occurrences of 2,, inside the boxed area, so it will not destroy any occurrences of ,2,3) to switch the positions of La and A. \ z__ Proof. Suppose Li and Li+l are .,li i,:ent 1 ivrs, and that fi > fi+l. Observe what happens if we switch the positions of Li and Li+l. Unless an occurrence of 7,~, has its /31 *.r in one of Li or Li+l, it will be preserved, just moved, so we need only worry about those occurrences of r2,,, whose /31 .r is contained in either Li or Li+l. Let y a + E C' be the number of elements of a to the left of Li, and let a be the number of increasing 2sequences in this range. Then, the number of occurrences of 7'2,, which have a /31 .vr in Li or Li+ is 'a ) a(ti1) +YA(ti). If we swap the 1'vrlengths, the number of these occurrences becomes a(tl') +aQc GO e+aQ $). Of course the sum of the first two terms remains the same. However, since fi > fi+1, the final term is certainly larger after the switch, by our technical lemma. It follows that if fi > fi+l, the permutation a cannot be 2,, maximal. Since this statement holds for all 1 < i < k 1, we can assume that if a is maximal, then its 1'.vrlengths are in increasing order. Lemma 4.4.6 If lal > 2 + 3, then we can assume that a begins with an a,r : .;,. of size at least 2 and ends with a 7.;,. of size at least 3. Proof. By Lemmas 4.4.3 and 4.4.5, if the last livr Lk is not at least as big as /, then there can be no occurrences of 7,,, in a, so a could not possibly be ~,,  optimal. Suppose a's initial antil*,vr A has less than two elements. Then, either A is empty or it is a single point. In either case, the first two elements of any occurrence of 7,23, must come separately from the first two I lr i of a, and so no occurrences of 72,, can occur entirely inside the first two blocks. It therefore does not decrease the number of occurrences of T,,, to exchange the first two blocks for an antiliv r of the same size. Lemma 4.4.7 We 1,,;.q assume that fk > a. Proof. Suppose that a > k, and suppose we remove the last element x of A and place a new element y at the beginning of Lk. We will show that there is a (strict) net increase in the number of occurrences of 2,,,, so that a cannot have been p2, ) maximal. First, we count the occurrences which are lost when we remove x: (1) If x is the first element of an occurrence, then the second element is in some Li and the 1'. r is in some Lj, j > i; the number of vi, this can happen is 1 (2) If x is a second element, then the first element must have been an earlier ele ment of A, and the lir could be in any Li; these occurrences are enumerated by (a i t) ( i= 1 Of course x is never in a 031,r because x E A. Next, we count the occurrences which are gained by putting y into Lk. In all new occurrences, y must be in a new /31 Ir, the other elements of which can be any 3 1 elements from Lk. The first two elements must come from one of three places: (1) Both can come from A; there are 2  such occurrences. (2) We can get one from A and one from another lir; there are (a 1) fki i 1 of these. (3) The two can come from two different 1.i, Li and Lj; there are (f k) > fi 1 of these. Hence the loss is i=1 l and the gain is (a t) k I (k )+(a 2 1) (k)+ ij 4k i1 l Naturally, to provide a contradiction to ,2,maximality of a, we will be showing that the gain must exceed the loss. It will later expedite our computation to consider separately the occurrences of 72,,, which have a /31 r in Lk: we can write the loss as the 4term sum (a 1) +(a l) +( ) +k 1 i Now, notice that for each j {1,..., k} we have 3j fj fj t < fk  because of our assumption that the lIris are ordered by increasing size. Now, we have k1 k1 (a 1) <(a 1) Qk Si 1 i 1) f/ ] t k1 t fi /G  and k1 fk k1 n fk E 4) E 4(i ) (by (4.3)) 1 f i k k f \ < 1t ( t) 1 tk1 because we are assuming also that fk < a. Hence, we are able to bound the sum of the first and last terms of (4.2) strictly below the first term of (4.1) because of our assumption that 3 > 3. That the third term of (4.2) is bounded below the third term of (4.1) is clear from our relation (4.3) on binomial coefficients. Finally, we may bound the second term of (4.2) by raw computation. By Lemma 4.4.6, we may assume that fk > > 3, so that a > 4 and thus a 1< (a 2). It follows now that <(aa 2). Itifollow(jnowk)hat ((a t) <(k < )( 3) k (k t) <(a 1)(a 2 k ) ^C;)^11 ( ( ) 1) (3 1) < (a 1) 4 (a 2) fk 2 A 1 We have thus bounded the loss strictly below the gain, so the change must have resulted in an increase in the number of occurrences of 7. In particular, to assume that a > k would be in contradiction to the ,2,,3maximality of a. We may assume, then, that Lk is at least as large as A. Finally, we may begin piecing together the information we have gathered about a to gain an important result. Theorem 4.4.8 For 7r e Sm, let gk (, n) = max {g (Tr, ) :: r E S, is of the form ALi... Lk and ILk > A}, 4.4 that is, maximize only over the permutations a having k nontrivial .;, ,. which riH;fy Lemma 4.4.7. Then, for each k > 2, we have k (72,3 n) < (gi 3 1n). Proof. It will suffice to show that gk (7,23 n) < g9ki (7 ,,) n for k > 2. Let a = AL1... Lk S,, be a permutation for which the expression (4.4) is maximized, and assume k > 2. In this context, we will not need the full strength of Lemma 4.4.3; simply write a = ALia*, as in Figure 44. We will show that replacing ALI with a single antiii r of size a + f1 does not decrease the number of occurrences of 7. Let 2 denote the number of occurrences of 72,3 that we lose when we remove A and L1, and let 6 be the number of occurrences we gain when we add in the ian til 'ir. Now, an occurrence of ,23,, is lost whenever it has at least one element in A U L1. In this case, either it can have its first two elements in A and a 1,r of size 3 in L1, its first two elements in A and a l1'i r of size 3 in a*, its first two elements from A and L1, respectively, and a l1v r of size 3 in o*, or a single element in A U L1 and the rest of the occurrence (i.e. an occurrence of r,,",) in a*. Accordingly, we have Z2 ( Q) Q +g (q 0> ,) r* (2) +alj (g ') [a i e] . When we place an .ili il ,ver of size a + 1I at the beginning of a*, we create two kinds of occurrences. First, any two elements in the new .il i'l ver, together with any lIr of size 3 in a*, will create an occurrence of 2,, Also, any element of the new Figure 44: Lessspecific structure of the permutation a 1 I I antil iv.r in concert with an occurrence of 7;,) in a* will create a new occurrence of 72,3 It follows that g (770,3) (a7) +g (71,) ) [a+l], so that the net gain is S(7,),*) [( a+ ) (a) ] (a) (f 1) 41 (the terms involving g (71, o*) cancel out). Now, in the first term of (4.5), the expression (a+1I) counts the number of vis to choose two elements from A U L1, and we subtract off the number af1 of v i, of choosing one from each set and also the number (a) of v i to choose both from A. If we do not choose both elements from A and we do not choose one from each set, we must choose two elements from L1, so the first term of (4.5) becomes Hence, we have We must show, then, that the expression (4.6) is ahv < nonnegative whenever u* is nonempty. Suppose first that il > a. Then, we have ( 2) [g r^' ) PJ Since (a) is ab v positive, it is sufficient to prove that g Tr,,3 ,o*) > ("); however, this inequality holds whenever a* has at least one 1v. r as large as L1, which follows from Lemma 4.4.5. In the case that ~1 < a, we know that (a) (%) < (Q) ( ) by our technical Lemma (4.4.1), so we have ,(^ ) 0 )(;)T f2) (l)[g(%g3,*) ()O which is nonnegative whenever a* has a 1ir at least as large as a. However, this fact is guaranteed by our adherance to permutations satisfying Lemma 4.4.7. We have thus reduced the number of nontrivial 1.. ri by one without decreasing the number of occurrences of ,,,, so that if a ,2,,3maximal permutation a has k nontrivial 1l.i~, it still can have no more occurrences of ,2~,3 than a 2,,,maximal permutation having k 1 nontrivial 11 r: the result follows. The main results of this section now follow as corollaries: Corollary 4.4.9 There is a pattern a E S, which maximizes the number of occurrences of 2,,, and which consists of a single .,l. i followed by a single 7., . Hence, the maximum number of occurrences of Tr2,, in a permutation in S, is g ,, = ax x 4.7 g (T"V^ n) x.E{o,...,n}\J 2 3 Proof. By Lemmas 4.4.3 and 4.4.7, we have g9 (r>, ) maxgk( ,, n). From here, it follows from Theorem 4.4.8 that g (r,,, n) = gi (', n), which is clearly (4.7). Corollary 4.4.10 The packing 1 ,.';/ of 2,,, is 3(+2) ( 2 ( )2( 3 2)3 S 2 0+2 0+2 Proof. Notice that we could easily rewrite the expression (4.7) as g (, n) max (tf.1(L(1 f> . The work of Price [14, Theorem 3.1] shows that a sequence (,) whose elements max imize the expression (4.8) approaches a constant that is, there is an ., iIl ically best ratio of the sizes of the 1 v.r and the antili, r of a. For large enough n, we approximate (, by the constant value (. It follows that 9 (r2,n v) p(r2 ) lim n (3n2) \#3+27 + 2 (t i3! Slim 2! (3+2)! (32) We may now maximize the expression (4.9). Noting first that (4.9) evaluates to 0 when is 0 or 1, we may maximize the expression over [ E [0,1] via elementary calculus; setting [ (L )10] = [2(1 _)" (1 ~ 1] = 0, we have 2(1 = f3, so that 2. The result follows. 4.5 Unimodality and LogConcavity In section 4.3, it was proven by an inductive technique that the 7,, optimal permutation of size 2n ahvli had exactly the same structure, independent of a, namely a single antiiv, r followed by a single 1lv. r of the same length. In general, if it can be proven that there is a r ," optimal permutation of size n having a 2block structure (a single .in il iver followed by a single li Vr), then the maximum number of occurrences of 7,,> in a permutation of length n is O In section 4.4, the existence of a 2block T ,,optimal permutation on [n] was proven in the case that a = 2 and 3 > 3, for n divisible by f + 2. Although we were then able to compute the packing density of T,,, via .,i~ ,! iI ic methods, we would like a stronger result characterizing the optimal layout of permutations of all lengths, that is, we would like to know the value of k achieving (4.10). To find it, we need to strengthen some general combinatorial concepts, that of unimodality and log concavity, to apply to more general functions. We will begin with the definitions of these words as they pertain to sequences. Definition 4.5.1 A sequence (an) of numbers is called unimodal there is some index no such that aT < an,+ for every n < no and aT > an,+ for every n > no. Definition 4.5.2 A sequence (an) of numbers is called logconcave if for every n we have aafnn+2 < a+2 It is a standard result that logconcavity is a stronger condition. Proposition 4.5.3 If (an) is logconcave, then it is unimodal. Proof. Suppose (au) is logconcave. Then, for each n we have an+2 < an+1 an+1 an that is, the ratio a+ is decreasing in n. In particular, if the ratio a" ever crosses below 1, it can never come back above it. In other words, once the sequence starts d,, ii.. it never starts increasing again, so the sequence is unimodal. To find the maximum value of a function whose domain and range are the reals, we wish to extend these ideas to more general functions. every x < y < z E [a, n 3]. Taking exponents, we have B(x)B(z) < B(y)2, so the result follows from the definition. Proposition 4.5.9 Let a,3 E Z+. The function B(x) 1. f;,'., in Lemma 4.5.8 is unimodal on [a, n 3] and achieves its maximum value on that interval at some point xo in the interval [ +/ , Proof. Since B(x) is a polynomial in x, it is continuous, so in particular it achieves a maximum value on the compact interval [a, n P]. That B is unimodal on this interval follows from the logconcavity result proven in Lemma 4.5.8. By the definition of B(x), we have B(x) (x)a (n x)3 B(x + 1) (x + 1), (n x 1)3 (x)cI(X a + 1) (n x)(n x 1)_1 (x + )(x)ai (n x 1)3(n x /3) (x a + )(n x) (x + 1)(n x 0x' It then follows that B(x) > B(x+l) if and only if (xa+l)(nx) > (x+l)(nx0), which, after some rearrangement of terms, turns out to happen exactly when x > "j. Thus the chord on the curve y B(x) from (x, B(x)) to (x + 1, B(x + 1)) a+/33 changes from having positive slope to having negative slope when x passes n"; so B(x) must change from increasing to decreasing somewhere inside the interval [ " n+ ). Since B is unimodal, the maximum value of B must be achieved in this interval. In the cases where we know that there is a 2block 7roptimal permutation of size n, the above characterization of the location of the maximum of this function gives us a good idea of the optimal size ratio of the two blocks. Corollary 4.5.10 Let n E N and suppose that we know there is a 7,3, optimal permutation of size n consisting of a single l,.: ~r followed by a single 7.';, ,. Then, n) \ ( +7) an+o /3n) for some integer ( with 3< < a. In particular, we have an+a \ //f'n+/3 \ If (a + 3) n, then r n) B ( }7 (a+) a+ Proof. Since there is a T,, optimal permutation of size n of the given structure, we know that g (,, n) is equal to the maximum value of B(k) for k c Z. We know the approximate location of the maximum value of B(k) for k c Z from Proposition 4.5.9, hence the first equality. The second inequality simply comes from the fact that (z)a and (z) are increasing functions of z. If (a + 3) n, then the unique integer in the interval [I +, ) is .a+" 4.6 Application to Larger Patterns We may now begin to reapply the inductive ideas applied to the pattern T, a in section 4.3, using more general 2block patterns as base cases. So that the induction flows smoothly, we will first need to prove the following technical lemma. Lemma 4.6.1 Let a, b, n,r c Z+ and suppose that r < a, n > a + b, and a rb. Let z rb and let n n (z + r) f = and g = a+b (a+b) (z+r) Then, af r = (a r)g and we have the following p.;,/':l/; among generalized binomial coefficients: (a af af (a r) F \ a r) ar ) Proof. That af r = (a r)g can be determined by elementary algebraic manip ulation. The equality (4.12) on integer binomial coefficients then has the following easy proof: choosing an aset A from a set S of size af and then choosing an rsubset from A is the same as first choosing an rsubset from S, then choosing the remaining ar elements from the remaining (ar)g. Since these two functions are polynomials, and they agree on an infinite number of points, they must agree everywhere. We can now begin our characterization of the similar behavior of similarly structured patterns. Theorem 4.6.2 Let a, 3, n E N, and suppose (a + 3) n. Suppose that there is known to be a r,, optimal permutation of size n consisting of a single r,:,.:r, followed by a single 7.';,. ,. Then, for every k E Z+, we have 3 \ n) a+3 a+f3 ,rwpka' ku Proof. Since (a + 3) n, that g T('ra>, n) > (ti3) (cg ) follows from constructing a permutation on [n] consisting of a single .inlil iver of size a and a single lvr of size 3." To prove the reverse inequality, we will proceed by induction on k. By Corollary 4.5.10, our base case k = 1 is covered, that is, we know g (r, ,n)= a+3 a+3 Assume that the theorem holds for 1,..., k 1. Suppose that ( a+n3 a+3+ ( 'rm ,n) k2)Qkp) for some 6 > 0, and let a E S, be a '7,rkk optimal permutation in S,. In each ocrecofinatra) () occurrence of in there are (ka k OCCUrencs of T ,,,. Consider an occurrence Of 'T{ka,kf3) in a, there ar\e ( a 1 occurrence To of T ,3 in a and occurrence r* of 'T,ra containing it. The elements of 7* \ To must form an occurrence of ;(k )1,(k )3 in the remaining n (a + 3) elements of a, so the number of occurrences of 7rk3> in a which can contain a single occurrence of ,>, is bounded above by g (T( )a,(k1)3) n ( (a + )). Since (a +P) n, of course (a + ) (n (a + 3)). Hence, we may apply the induction hypothesis to get (a(n (a+3))\ 13(n(a+3)) (k 1)a (k 1) Now, applying Lemma 4.6.1 (first using {a = ka, b = k3,r = a}, then {a = k3, b ka, r = 3}), we have (ka (k 3) ann n r a >18 _a+P a+P+ q n ka [ k + (k1)a \ (k1)3 ( an k3) (aQ\8n for some E > 0, which contradicts the result of Corollary 4.5.10. Notice that Theorem 4.6.2 also characterizes the structure of a To ,k3>optimal permutation in the cases where it applies. Corollary 4.6.3 Suppose (a + 3) n. If there is a T,.,) permutation g e S,, consisting of a single .1.,. followed by a single 1. then is in fact Tk,,3> optimal for every k E N. If we drop the divisibility condition on n, we can still get an upper bound on g (7,>, ) by similar means. Theorem 4.6.4 Let a, 3, n E N, and let x be a real number such that xa, xp E N. Suppose there is known to be a T,3> optimal permutation of size n consisting of a single ,,.:nl.. r followed by a single 1.,. and we know (an+a \ (3n+13 g~r ^n < )a + )* Then for every r E N we have ( an+a \ 3n+3 an) < a+ } a+3 (x + r)c x( + r); Proof. The proof is done here by induction on r; we assume the case r = 0 in the hypotheses. The induction step is done by a similar approach to that of Theorem 4.6.2. Assume the theorem holds for 0,..., r 1, and suppose that / an+a \ /3n+p/3 i r(r") x + r)) (x + r)( for some 6 > 0. Let a be a r,( +,).,( +,r, optimal permutation on [n]. As before, we now count the number of occurrences of 7T,,) in a. Certainly in any occurrence of (~+,x. ,( ,r) in o, there are ((x+r)) ( ) occurrences of 7,>. Let To be a specific occurrence of 7;r,, in a. For any occurrence T* of 7(,+,> ,(a+,>,> containing ro, the remaining (x + r 1)((a + 3) entries of T* must form an occurrence of '( +,r 1)a,(x+r 1) in the remaining (n (a + /)) entries of a. Hence the number of occurrences of ra(x+, ,(+,),, containing To is bounded above by g (r(, r1), 1),, a (a + 3.)) Now, by induction, we know that so we have g (7'rae a) Since an+a e a+li equality ((n(a+ 3)+a 1 3(n_ (a+3))+3 + ar+3 a+/3 (x+r )a (x + r 1)/ ana 3n+ ((x+r)a) (x+r)3) a(n+a /3n+/3 ( r)a a+ ) (x + r) ( + r)x > a siia a+e3 t a+13 (m+r1)a (+r1)/3 / a a+3p ( an+a / / a a+ r (x + r)a a (an+a\ a(n(a+3))+a a+P3 + r 3) a (x + r 1)a) g (r(x r 1)a )3 ,T (a + 3))< holds. Similarly, since + = 3 (n(a++f))+3, we can show that 3n+3 P n+P3 3(n (a+))+3 a+P + a+p a+ (x + r)/3A (x + r 1>)/3 It now follows from (4.13) that an+ a 1/3n+ 13 for some E > 0, which contradicts the results of Corollary 4.5.10. The result follows. 4.7 Concrete Results In the case that a = 2, the 2block structure of a ro,,>optimal permutation was proven in Section 4.4, so we may now use the more general results of the previous sections to compute the packing densities of a specific class of patterns. In particular, Corollary 4.5.10 gives us exactly the characterization we wanted of an infinite family of T2,3 optimal permutations, which will provide an essential base case to which we can apply Theorem 4.6.4. Corollary 4.7.1 Suppose 3 > 2 and (3 + 2) n. Then, there is a 2,3 optimal permutation c2,8(n) on [n] consisting of a single ,,i.:,, of length 27, followed by a 7.;,. of length 73. It follows that n(, ,) o ') (2 ) (2+) 2+ Theorem 4.7.2 Suppose 3 > a > 2 and a 2/3. Then, for every n divisible by (2 + 2), we have /an+a /Pn+P ^*" aX 0 Proof. By Theorem 4.6.4 (applied with x = 1 and x = ), we need only prove that 2n+2 /n+ 3n+3 3n+P 2n 3n We know that g (7 ,,,, = (2 (23) from Corollary 4.7.1; to prove the other ( 3n 3n inequality, we will prove that g (T3, n) < ( ) (3 ) Let a 3. Then, a 20 means that 3 3. C'! ... n divisible by 2 + j3, and let a be a 3 optimal permutation of size n. Let r = y = 2r, and suppose that S73,3, n) 3+3> (j 3+(i3 + for some 6 > 0. Let a S, be 7 3,,optimal. Certainly any occurrence of 73, in a must contain exactly 3 () occurrences of 72, Let To be an occurrence of 2,Y, in a. Since 73,, is 1 ,.'. i, Theorem 2.2.2 allows us to assume a is 1,i,. 1, so let x be the location of the first element in the 1..r of a that contains the li. r of size y in 7o. Now, consider an occurrence * of 73,,3 that contains 7o. It is clear that none of the first three elements of 7* can be to the (weak) right of x and that none of the last f elements of 7* can be to the (strict) left of x (see Figure 45). It follows that the number of occurrences of 73,, in a that can contain To is bounded above by the number (3) nsy)+l), which by Corollary 4.5.102 is bounded above by (ny2 \ /(3y)(ny2) 1+(3y) 1 +(3y) 1 0y ) Since this number is independent of our choice of T7, it follows that no occurrence of ny2 (3y)(ny2) ,7 can be contained in more than +'(32y) ( 1+ ~ y) ) occurrences of 73,, in a, so 2 Since we assume (2+y) = (2r+2), which divides n, we also know that 1+(/y) (r + 1), which divides (n 2r 2) = (n y 2). The reason the case a = 3 is spe cial is because in this case, the size of 7* \ o is exactly half the size of 7o, so the divis ibility works out perfectly to satisfy the divisibility hypothesis of Proposition 4.5.9. Figure 45: An occurrence TO of in an unknown permutation a is shown for y=4. Notice that we can only get an occurrence of T3,6) containing TO by adding elements to the shaded regions one to the left region and two to the right. The only assumption we make about a is that it is 1 ,1.v . we have 3 ( 3 ) 3 n 3' N ] 6 g(T2,Y 17) > ( ) (3 (y) 3+/2) 3 \3 / 1 / \ 3y / 3n /3n > +3 /+3 + 2n ) (yn_ v+2 y+2 + for some E > 0, where the second inequality comes from Lemma 4.6.1 (first with {a = 3, b = 3, k = 2} and then with {a = b = 3, k = y}). However, since (2 + y) n, ( n y this statement contradicts the result of Corollary 4.7.1 that g (2r,,n (In2) (vY2) because by definition g (T2,y, a) is bounded above by g (T,,, n). The result for a = 3 follows. We may now compute the packing density of 7,,> as a corollary. Corollary 4.7.3 Suppose 3 > a > 2 and a 20/. Then, the packing /. ,./'/ of the pattern T,,) is p70 a ) + ) a +4.14 Proof. By letting ua,3(n) be the permutation on [n] consisting of an antiliJ r of length L~a followed by a li r of length ,3 we ensure that we have 0 * 59 9 ( ) (L3 ) ( N33). Hence, by Theorem 4.7.2, we have (L 8) ) n a+/37 \a+/37 \a+/37 for every n divisible by 2 + 2. Certainly the limits of our two bounds are the same, so the result (4.14) follows from the squeeze theorem. CHAPTER 5 EXTENSIONS TO MORE GENERAL PROBLEMS The permutation packing problem has been generalized in several directions. Here we provide a brief survey of some related packing problems that people have been working on since the inception of packing density, and the progress that has been made along those lines. 5.1 On Unlayered Patterns In general, the case of url liv ri I patterns is largely unknown territory, as we are left without the resources of Theorem 2.2.2. However, we may achieve upper and lower bounds on packing densities of u:rl li, rI ,t patterns by computational means. Albert et al [1] proved the following bounds for 7 = 2413. Proposition 5.1.1 We have 2a < p(2413) < 2 The structure of the permutation achieving this lower bound is interesting in and of itself, as it is of a fractal nature. Start with a roptimal permutation a E Sk. Replace each element i with a copy of a; recur as shown in Figure 51. This process creates an infinite family of permutations by which we may calculate a lower bound on p(T). In general, we can get an upper and a lower bound on p(T) by computational means. Figure 51: The first three iterations of the algorithm applied to 7r = 3142. Proposition 5.1.2 For every pattern 7 e Sm and r > m, we have mn!g (, r) < 7) m!g(, r) 5.1 r <r (r)m, Proof. The upper bound comes directly from Galvin's theorem that the sequence g(',n) decreases to its limit. To get the lower bound, we construct an "infinite permutation" that will act like a family of permutations. Start with a Toptimal permutation 7 E S,, and let ao = 7. In each step, construct a permutation rk+1 of size rk+1 by setting (1) (2) (r) k+1 k k k where each aj is isomorphic to rk and every element of aV) is greater than every element of j) if and only if 7(i) > 7(j). In other words, we replace each element of 7 with a smaller copy of Jk, as in Figure 51. What we end up with is a structure which is divided up into r regions, each of which looks like a scaleddown 'iv of the whole permutation. Consider the permutation ak of size rk that is built in the kth step. Let Pk be the probability that a randomly selected msubset of [rk] is an occurrence of 7 in k*. There are two natural vv to get occurrences of 7 within this structure. We can choose m elements from the same subpattern with probability (() which is ( 4). The chosen subset will be an occurrence of 7 with probability pk . There are r such subpatterns, so the total probability of this scenario is (y1i )) Pk( K Alternatively, we can choose the m elements each from different subpatterns so that they form an occurrence of 7 if and only if the external ordering of the subpatterns is an occurrence of 7 in ao. There are g(T, r) such subpatterns, so the total probability of this scenario is (,) m(,r) total probability of this scenario is ( 1)g > rmg (Ir;) In some cases there may be other v,v to get an occurrence of r in ak, but we can at least get a lower bound on pk. We have ( 1 oF 1 ) m!g(r, r) Pk (r 1 0 (rpkl P 1 + rpm Letting v = lim inf pk, we have vr + m!g(Tr, r) > rr which leads to > m!g(r, r) Since v = liminf (k), we know p(7) > v. Note also that these particular computational bounds can be arbitrarily close to the packing density, as the limit in r of both sides of (5.1) is p(T). It is the de termination of g(r, r) that makes the computation difficult, as it requires examining basically every permutation of size r and counting occurrences of r (there is no rea son to examine permutations which avoid r, but the solution [13] of the StanleyWilf conjecture tells us that in the long run there are not too many of these). Applying the above theorem to the case 7 = 2413 and using C code slightly modified from that of Johnson and Barker [10], we come upon the computational result that the value of g(2413, 12) = 88 is achieved by the pattern 5 4 712 11 3 10 2 16 9 8 of length 12, and we obtain the following minor advancement of Proposition 5.1.1: Corollary 5.1.3 We have 16 8 7 < p(2413) <  157 45 Although there is no theoretical advance, the gap between the bounds is narrowed to approximately i1' of what was previously known. 5.2 Packing Sets of Patterns The idea of packing a set of patterns, rather than just an individual pattern, was first approached by Albert et al [1]. We will shift the notation used by the authors to resemble our earlier choice of notation for single patterns. Notation 5.2.1 Let II be a set of patterns of the same length m, and let a E S, for some n > m. Let g(II, a) be the number of occurrences in a of members of I. Similarly to our previous notation, we let g(H, n) be the maximum value of g(H, a) over all a e S,. Accordingly, the permutations a E S, for which g(I, a) = g(I, n) are called IIoptimal. A result similar to that of Galvin is also true in this more general case: Proposition 5.2.2 ([1, Proposition 1.1]) The sequence (L is nonin n>rn creasing in n. We may therefore define the packing density of a set of patterns of the same length in the same way as we defined it for an individual pattern. Definition 5.2.3 The packing density of the set I is the limit pmn^9^ p(II) lihma(I ) Stromquist's Theorem (2.2.2) also has an analogue in this case, even for the more general situation where the elements of I may be repeated: Theorem 5.2.4 ([1, Theorem 2.2]) Let I be a multiset of 7.';;' .1 permutations. Among the Ioptimal permutations of each length, there is one that is 7.';/. ..l. Furthermore, if all the 7.'; ,. of every element of I have size greater than 1, then every Ioptimal permutation is 17.I,. ..l. From here, the packing behavior of a few sets was explored [1]: Theorem 5.2.5 ([1, Theorem 2.5]) Let II {71,2, where w1 is l.;, .. of ';," (k, f) and 72 is 1,,,, ,, ,I of type (f, k). Then, every IIoptimal permutation of length at least k + has il. //11 two 17.I, . In particular, the authors were then able to compute p({132, 213}) = . 5.3 Allowing Repeated Letters A similar version of the packing problem can be stated when the permutations involved are allowed to contain repeated letters. Most of the known results in this area were proven by Burstein, Hast6, and Mansour [7] and Barton [2]. In this context, the definition of occurrences are the same, that is, S is an occurrence of a pattern 7 in a word a if a [s is orderisomporphic to 7, where we allow equality as an order. For example, the sets {1, 3, 4}, {2, 3, 4}, and {2, 5, 6} are occurrences of the pattern 122 in the word 213322. Notation 5.3.1 Let II be a collection on nonisomorphic words of length m. For a word a E [k]', let g(II, a) be the number of occurrences of an element of I in a. Let 6(n, k, n) =max {g i .a [k]"}. Since we still define an occurrence as a subset, the total number of occurrences of nonisomorphic members of a set II in a word of size n is still bounded above by (n), so by examining the .i''iiil 'I .ic behavior of the function 6(I, k, n), we are accomplishing the same goal that we did with nonrepeating patterns. Theorem 5.3.2 ([2, Theorem 3]) For II a collection of patterns of length m, we have lim6(II, n, n) = limlim (II, k, n). n k n Definition 5.3.3 The packing density of the set II, denoted p(II), is the common value of the two limits above. To show that this new definition of packing density concurs with our old one, it was shown by Burstein et al [7] that if the members of II are permutations (without repeated letters), then the number we get will be the same; that is, it does not help us to pack permutations into words with repeated letters. Theorem 5.3.4 ([7, Theorem 1.2]) Let I C S, be a set of permutations. Then, marfg(fl, 7) :: 7 G[n]nT p( n) m ) Smax{g(H, a) :: ac S,} rn (fl) that is, the packing 1 ,;'.:// of I on words is equal to that on permutations. It was next proven [7] that nondecreasing patterns, i.e. patterns whose elements are in nondecreasing order, have similar behavior to that of lii I permutations, and that the blocks of the same element in this case 1 li, a similar role to that of Theorem 5.3.5 ([7, Theorem 2.1]) Let H E [1]' be a set of nondecreasing patterns of the form 7 = 1a1()22() .. w). For each 7r II, let ir be the I,;., ', I permutation of size m with ,.;, , (al(7r), a2(7r),.. aG()), and let U= {w :: 7r II}. Then, P(n) = P(0). We may also define what it means for a more general pattern to be 1 li ,. I1. Definition 5.3.6 A pattern 7r is called layered if it can decomposed into a sequence of blocks (1'.r ) so that within each 1I.;;, the elements are in nonincreasing order, and between 1.,;, , the elements are in increasing order. A theorem similar to that of Stromquist has been conjectured but not proven: Conjecture 5.3.7 ([7]) Let I be a set of '.,;, .1 patterns. Among the set of Hoptimal words in [k]", there is one which is 1i.,, .1. To prove the conjecture for a subclass of 1 li1 patterns, Barton introduced the notion of clumpy patterns. Definition 5.3.8 Let E be an alphabet. A word 7r E E is called clumpy if for ,;, letter x E E, all occurrences of x in 7~ occur together. The following analogue of Stromquist's theorem was proven by Barton [2] for clumpy patterns: Theorem 5.3.9 ([2, Theorem 6]) Let I be a set of clumpy patterns. Among Uoptimal words in [k]', there is one which is clumpy. 66 Finally, the packing density of a specific class of patterns was established as a function of packing densities of some 1'. 1i permutations. Theorem 5.3.10 ([2, Corollary 14]) Let 7r be the pattern P3q2r, and let 7r1 be the I,;/;, ,, permutation pattern of ';pI' (p, q + r) and 7r2 the 1';,I ,, Il permutation pattern of type (q, r). Then, P () =P (1)p( 72). REFERENCES [1] M. H. Albert, M. D. Atkinson, C. C. Handley, D. A. Holton, and W. Stromquist. On packing densities of permutations. Electronic Journal of Combinatorics, 9:#R5, 2002. [2] R. W. Barton. Packing densities of patterns. Electronic Journal of Combina torics, 11:#R80, 2004. [3] M. B6na. Symmetry and unimodality in tstack sortable permutations. J. Combin. Theory Ser. A, 98(1):201209, 2002. [4] M. B6na. Combinatorics of Permutations. CRC Press, Boca Raton, FL, 2004. [5] M. B6na. The limit of a StanleyWilf sequence is not ah,v rational, and 11 1. patterns beat monotone patterns. J. Combin. Ti, ..", Ser. A, 2005. to appear. [6] L. Breiman. P, '1..1'.:.:l;, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1992. [7] A. Burstein, P. Hast6, and T. Mansour. Packing patterns into words. Electronic Journal of Combinatorics, 9(2):#R20, 2003. [8] P. A. Hast6. The packing density of other 1 1,i 1. permutations. Electronic Journal of Combinatorics, 9(2):#R1, 2002. [9] M. Hildebrand, B. E. Sagan, and V. R. Vatter. Bounding quantities related to the packing density of 1( + 1)... 2. Advances in Applied Mathematics, (33):633 653, 2004. [10] J. Johnson and C. Barker. Revisiting lexicographical permuta tion methods. Webpublished Manuscript, 1998. Accessed at http://www.cs.byuh.edu/research/johnson/permutation/ on January 8, 2004. [11] M. Klazar. The FiirediHajnal conjecture implies the StanleyWilf conjecture. Formal Power Series and Algebraic Combinatorics, Springer V, i1 Berlin, pages 250 255, 2000. [12] D. E. Knuth. The Art of Computer P, .,'ri,,,,,:,.' j volume 1. AddisonWesley, Reading, PA, 1968. [13] A. Marcus and G. Tardos. Excluded permutation matrices and the StanleyWilf conjecture. J. Combin. Th, , y Ser. A, 107(1):153160, 2004. 14] A. Price. Packing Densities of L';' .1 Patterns. PhD thesis, University of Pennsylvania, 1997. 15] B. E. Sagan. The Si, ii, iii Group: Representations, Combinatorial Algorithms, and Siiiiiin, I,'. Functions. Number 203 in Graduate Texts in Mathematics. SpringerV. i1 . New York, NY, 2001, 2nd edition. [16] R. P. Stanley. Enumerative Combinatorics Volume 1. Number 49 in Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge UK, 1997. 17] R. P. Stanley. Enumerative Combinatorics Volume 2. Number 62 in Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge UK, 1999. [18] W. Stromquist (Bryn Mawr College). Packing 11,,. 1 posets into posets. Unpublished Typescript, 1993. [19] R. Tarjan. Sorting using networks of queues and stacks. Journal of the Association for CornrI.:,il Machinery, 19:341346, 1972. BIOGRAPHICAL SKETCH Dan Warren was born on April 10, 1978, in Thomasville, Georgia. On December 18, 1999, he graduated summa cum laude from the University of Georgia with a Bachelor of Science in mathematics and married Jennifer Mathews of Pavo, Georgia. He enrolled in the mathematics program at the University of Florida in August of 2000 and received a Master of Science in mathematics in August of 2002. 