RESTRICTED PERMUTATIONS AND POLYGONS. Ghassan Firro and Toufik Mansour Department of Mathematics, University of Haifa, Haifa, Israel

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1 RESTRICTED PERMUTATIONS AND POLYGONS Ghassan Firro and Toufik Mansour Department of Mathematics, University of Haifa, 905 Haifa, Israel abstract Several authors have examined connections among restricted permutations and different combinatorial structures In this paper we establish a bijection between the set of permutations π which avoid and the set of odd-dissection convex polygons, where a permutation avoids ab c if there are no i < j < k such that π i π j π k is order-isomorphic to abc We also exhibit bijections between the set of permutations that avoid (or ) and the set of odd-dissection convex polygons Using tools developed to prove these results, we give enumerations and generating functions for permutations which avoid and certain additional patterns Extended abstract Classical patterns Let [n] = {,,, n} and denote by S n the set of permutations of [n] We shall view permutations in S n as words π = π π π n We denote by S the set of all permutations of all sizes (including the empty permutation ɛ, that is, the permutation of length 0), that is, S = n 0 S n The reduced form of a permutation σ on a set {j, j,, j k }, where j < j < < j k is a permutation obtained by renaming the letters of the permutation σ so that j i is renamed i for all i {,, k} For example, the reduced forms of the permutations 497 and 974 are 4 and 4, respectively Definition For k n, we say that a permutation σ S n has an occurrence of the pattern φ S k if there exist i < i < < i k n such that the reduced form of σ(i )σ(i ) σ(i k ) is φ We denote the number of occurrences of the pattern φ in the permutation σ by φ(σ) We say that a permutation π avoids a pattern φ, or is φ-avoiding, if φ(π) = 0 For example, let π = 8754, φ = 4 and θ = 4 Then it is easy to see that π avoids φ, and contains exactly one occurrence of θ, that is π does not avoid θ The set of all φ-avoiding permutations in S n is denoted by S n (φ) For any set T of patterns, we let S n (T ) = φ T S n The first explicit result seems to be Hammersley s enumeration of S n () in [5] In [, Ch ] and [, Ch 54] Knuth shows that for any τ S, we have S n (τ) = C n, where C n is the nth Catalan number given by C n = n+( n ) n (see [9, Sequence A00008]) Other authors considered restricted permutations in the 970s and early 980s (see, for example, [4], [5], and []), but the first systematic study was not undertaken until 985, when Simion and Schmidt [7] solved the enumeration problem for every subset of S Currently, there exist more than two hundred papers on this subject (see [8])

2 RESTRICTED PERMUTATIONS AND POLYGONS Generalized patterns In [] Babson and Steingrímsson introduced generalized permutation patterns that add the requirement that two adjacent letters in a pattern must be adjacent in the permutation In order to avoid confusion we write a classical pattern, say, as --, and if we write, say -, then we mean that if this pattern occurs in the permutation, then the letters in the permutation that correspond to and are adjacent Let us give a formal definition of a generalized pattern Definition A generalized pattern of length k is a word φ = φ x φ x k φ k, where φ φ φ k S k, and for j =,,, k, x j is either the empty string ɛ or a dash - If x j = - then in the definition of an occurrence of a classical pattern we require i j i j +, otherwise we require i j = i j + For example, the permutation π = 45 has two occurrences of the pattern --4, namely -4- and -4-5 A number of interesting results on generalized patterns were obtained in [] Relations to several well studied combinatorial structures, such as set partitions (see []), Dyck paths (see []), Motzkin paths (see [4]) and involutions (see [8]) were shown there As in the paper by Simion and Schmidt [7] dealing with the classical patterns, Claesson [], Claesson and Mansour [] considered a number of cases where permutations have to avoid two or more generalized patterns simultaneously In [7] Kitaev gave either an explicit formula or a recursive formula for almost all cases of simultaneous avoidance of more than two generalized patterns of length three with no dashes (see also [9, 0]) Distanced patterns In this section we give a uniform language to studying the classical pattern problem (see Definition ) and generalized pattern problem (see Definition ) in terms of the d-pattern problem Definition A distanced-pattern (or d-pattern) of length k is a pair (φ, d) where φ S k and d is a word d = d x dx dx k k such that d j 0 for j =,,, k, and x j is either the empty string ɛ, a minus - sign, or a plus + sign If x j = ɛ (resp x j = +, x j = - ) then in the definition of an occurrence of a classical pattern we require i j i j = d j (resp i j i j d j, i j i j d j ) For example, if π = 4578 S 8 then it contains Φ = (, ), eg π π 5 π 7 = 48 with distance d =, it contains Θ = (, 0 + ), eg π π π = 4 and π π π 8 = 4, and it contains Γ = (, 0 ), eg π π π = 4 and π 5 π π 7 = 8 As a remark, our Definition generalizes the classical and generalized definitions of patterns For example, avoiding the classical pattern 4 is the same as avoiding the d-pattern (4, ) and avoiding the generalized pattern -4- is the same as avoiding the pattern (4, ) The following two examples connect the d-pattern avoidance problem to binomial coefficients and Fibonacci numbers Example 4 Let d be any nonnegative integer number Then it can shown that #S (d+)n+l ((, d)) = l for all n 0 and 0 l d j=0 ( (d+ j)n+l j ) d n+ j=l ( (d+ j)n ) ((d + )n + l)! n = (n + )! l n! d+ l, Example 5 For any n 0, #S n (, + ) = F n+, where F n+ is the (n+)-st Fibonacci number To see that, let a n = #S n (, + ) For every permutation π in S n (, + ) there are two possibilities: the entry can be either the last (the n-th) element of π, or the (n )-st element of π In the later case the entry must be the last element of π Therefore, in the first case we have a n permutations, and in the second case we have a n permutations, hence a n = a n + a n Observing that a 0 = a =, we conclude that a n = F n+, as claimed

3 RESTRICTED PERMUTATIONS AND POLYGONS Define an odd-dissection convex polygon permutation or odd-dissection gon permutation (or ODPpermutation) π to be a permutation in S n that avoid the d-pattern, where we denote the d-pattern (abc, ) by ab c For example, there are exactly twenty ODP-permutations of length 4 We denote the set of all ODP-permutations in S n by O n The main reason for the term ODP-permutation is that the cardinality of the set O n is given by number of odd-dissections of a (n + )-gon The main results of this paper can be formulated as follows Let G n be a convex n-gon in the plane R with vertices labeled,,, n and edges,,, (n )n, n Figure The set 4 A dissection of G n is a partition of connecting vertices of G n into k polygons G,, G k by noncrossing diagonals of G n An odd-dissection of G n is a dissection G,, G k of G n such that G i is not a m-gon (m > ) for all i =,, k We denote the set of all odd-dissections of a given convex (n + )-gon by n (see Figure for the case n = 4) Observe that every odd-dissection G n has one of two forms: () The vertices and n + are connected by straight line segments to the same vertex i, and () The two vertices and n + of G are not connected by a straight line segment to the same vertex Theorem (i) There exists a bijection Θ between O n and n (ii) There exists a bijection between the set of -avoiding permutations in S n and n (iii) There exists a bijection between the set of -avoiding permutations in S n and n Let F (x) = n 0 #O nx n, then Theorem (i) gives F (x) = + xf (x) + x F (x)(f (x) ), and the values of the corresponding sequence are,,,, 0, 7, 4, 05, 400, 094, 5758, 708, 98 for n = 0,,, (see [9, Sequence A0494]) To find an exact formula for the number of ODP-permutations on [n], let p(x; α) = αx(p(x; α) + ) ( + xp(x; α)) Clearly, p(x; ) = F (x) On the other hand, by using the Lagrange inversion formula we get that p(x; α) = n n j=0 n ( )( n n ) j j + x n j α n Therefore, the generating function F (x) can be presented as F (x) = + ( )( ) n k n k x n n k n k k n k 0

4 4 RESTRICTED PERMUTATIONS AND POLYGONS Hence, we have the following result Corollary 7 For all n, the number of ODP-permutations, -avoiding permutations, - avoiding permutations in S n is given by ( )( ) n k n k n k n k k k 0 An another application of the bijection Θ to give the generating functions for several statistics in ODP-permutations For a permutation π, denote by τ k (π) the number of occurrences of the classical pattern τ k = (k + )(k) (in other words, τ k = ( (k + )(k), )), for any k For an odd-dissection n-gon G with partition into k-polygons G, G,, G k, denote by p k (G) the number of polygons G i with k + vertices These statistics can be characterized in terms of pattern avoidance as follows Lemma 8 Let π O n and G = Θ(π) Then τ l (π) = 0 if and only if p l+ (G) = 0, for any l Let F (t; x, x, ) be the generating function n 0 (t n ) π On l xτ l(π) l By Lemma 8 together with Theorem we have that the generating function F (t; x, x, ) satisfies F (t; x, x, ) = + tf (t; x, x, ) + t F (t; x, x, )(F (tx ; x x, x x, ) ), which is equivalent to () F (t; x, x, ) = + ( t) 4t F (tx ; x x, x x, ) By applying () repeatedly and in each step performing some rather tedious algebraic manipulations we get Corollary 9 The generating function for the number of ODP-permutations in S n is given by F (x;,, ) = + ( x) 8x As an another application of (), we have the following result + ( x) 8x Corollary 0 The generating function for the number of ODP-permutations that avoid τ l is given by H l (x) where H l (x) = + ( x) 4x H l (x) with H 0 (x) = For example, the generating function for the number of ODP-permutations that avoid the classical pattern (which equals the number of -avoiding permutations; see []) is given by H (x) = +, the generating function for the Catalan numbers Also, the generating function for the 4x number of ODP-permutations that avoid the classical pattern 54 is given by H (x) = + ( x) 8x + 4x

5 RESTRICTED PERMUTATIONS AND POLYGONS 5 References [] E Babson and E Steingrimsson, Generalized permutation patterns and a classification of the Mahonian statistics, Séminaire Loth de Combin 44 (000) Articale B44b [] A Claesson, Generalized pattern avoidance, Europ J of Combin (00), 9 97 [] A Claesson and T Mansour, Enumerating permutations avoiding a pair of Babson-Steingrímsson patterns, Ars Combin, to appear, preprint mathco/ [4] R Donaghey and LW Shapiro, Motzkin Numbers J Combin Theory Ser A (977), 9-0 [5] J M Hammersley, A few seedlings of research, Proceedings Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume, Berkeley/Los Angeles, 97, University of California Press, [] GH Hardy and EM Wright, Partitions, An Introduction to the Theory of Numbers, 5th ed Oxford, England: Clarendon Press, 979, 7-9 [7] S Kitaev, Multi-avoidance of generalised patterns, Discr Math 0 (00), [8] S Kitaev and T Mansour, Survey on certain pattern problems, preprint in wwwmsukyedu/math MAreport/indexhtml [9] S Kitaev and T Mansour, Simultaneous avoidance of generalized patterns, Ars Combin, to appear [0] S Kitaev and T Mansour, On multi-avoidance of generalized patterns, Ars Combin, to appear [] DE Knuth, The art of computer programming, Volume, Fundamental algorithms, Addison Wesley, Reading, Massachusetts, 97 [] DE Knuth, The art of computer programming, Volume, Sorting and Searching, Addison Wesley, Reading, Massachusetts, 97 [] SG Mohanty, Lattice Path Counting and Applications New York, Academic Press, 979 [4] D G Rogers, Ascending sequences in permutations, Discr Math (978) 5 40 [5] D Rotem, On a correspondence between binary trees and a certain type of permutation, Info Proc Letters 4 (975) 58 [] D Rotem, Stack sortable permutations, Discr Math (98) 85 9 [7] R Simion and F Schmidt, Restricted permutations, Europ J Combin (985) 8 40 [8] S Skiena, Involutions, Implementing Discrete Mathematics, Combinatorics and Graph Theory with Mathematica, Reading, MA, Addison-Wesley, (990) - [9] N Sloane and S Plouffe, The Encyclopedia of Integer Sequencess, Academic Press, New York, 995