RESTRICTED PERMUTATIONS AND POLYGONS Ghassan Firro and Toufik Mansour Department of Mathematics, University of Haifa, 905 Haifa, Israel {gferro,toufik}@mathhaifaacil 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])
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, 0 + 0 + 0 + ) and avoiding the generalized pattern -4- is the same as avoiding the pattern (4, 0 + 00 + ) 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
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, 0 + + ) 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 5 4 5 4 5 4 5 4 5 4 5 4 5 4 5 4 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 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), 0 + 0 + )), 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
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/007044 [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, 45 94 [] 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), 89 00 [8] S Kitaev and T Mansour, Survey on certain pattern problems, preprint 00-09 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