Restricted Permutations and Polygons

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 1322 and the set of odd-dissection convex polygons, where a permutation avoids ab2c if there are no i < j < k − 1 such that πiπjπk is order-isomorphic to abc. We also exhibit bijections between the set of permutations that avoid 1223 (or 2123) 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 1322 and certain additional patterns. 1. Extended abstract 1.1. Classical patterns. Let [n] = {1, 2, . . . , n} and denote by Sn the set of permutations of [n]. We shall view permutations in Sn as words π = π1π2 . . . πn. We denote by S the set of all permutations of all sizes (including the empty permutation , that is, the permutation of length 0), that is, S = ∪n≥0Sn. The reduced form of a permutation σ on a set {j1, j2, . . . , jk}, where j1 < j2 < · · · < jk is a permutation obtained by renaming the letters of the permutation σ so that ji is renamed i for all i ∈ {1, . . . , k}. For example, the reduced forms of the permutations 4973 and 1974 are 2431 and 1432, respectively. Definition 1. For k ≤ n, we say that a permutation σ ∈ Sn has an occurrence of the pattern φ ∈ Sk if there exist 1 ≤ i1 < i2 < · · · < ik ≤ n such that the reduced form of σ(i1)σ(i2) . . . σ(ik) 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 = 83176254, φ = 1234 and θ = 1243. 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 Sn is denoted by Sn(φ). For any set T of patterns, we let Sn(T ) = ∩φ∈T Sn. The first explicit result seems to be Hammersley’s enumeration of Sn(321) in [5]. In [11, Ch. 2.2.1] and [12, Ch. 5.1.4] Knuth shows that for any τ ∈ S3, we have |Sn(τ)| = Cn, where Cn is the nth Catalan number given by Cn = 1 n+1 ( 2n n ) (see [19, Sequence A000108]). Other authors considered restricted permutations in the 1970s and early 1980s (see, for example, [14], [15], and [16]), but the first systematic study was not undertaken until 1985, when Simion and Schmidt [17] solved the enumeration problem for every subset of S3. Currently, there exist more than two hundred papers on this subject (see [8]). 1 2 RESTRICTED PERMUTATIONS AND POLYGONS 1.2. Generalized patterns. In [1] Babson and Steingŕımsson 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 231, as 2-3-1, and if we write, say 2-31, then we mean that if this pattern occurs in the permutation, then the letters in the permutation that correspond to 3 and 1 are adjacent. Let us give a formal definition of a generalized pattern. Definition 2. A generalized pattern of length k is a word φ = φ1x1φ2 . . . xk−1φk, where φ1φ2 . . . φk ∈ Sk, and for j = 1, 2, . . . , k − 1, xj is either the empty string or a dash “-”. If xj =”-” then in the definition of an occurrence of a classical pattern we require ij ≥ ij−1 + 1, otherwise we require ij = ij−1 + 1. For example, the permutation π = 314265 has two occurrences of the pattern 2-31-4, namely 3-42-6 and 3-42-5. A number of interesting results on generalized patterns were obtained in [2]. Relations to several well studied combinatorial structures, such as set partitions (see [6]), Dyck paths (see [13]), Motzkin paths (see [4]) and involutions (see [18]) were shown there. As in the paper by Simion and Schmidt [17] dealing with the classical patterns, Claesson [2], Claesson and Mansour [3] 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, 10]). 1.3. Distanced patterns. In this section we give a uniform language to studying the classical pattern problem (see Definition 1) and generalized pattern problem (see Definition 2) in terms of the d-pattern problem. Definition 3. A distanced-pattern (or d-pattern) of length k is a pair (φ,d) where φ ∈ Sk and d is a word d = d1 1 d x2 2 . . . d xk−1 k−1 such that dj ≥ 0 for j = 1, 2, . . . , k − 1, and xj is either the empty string , a minus “-” sign, or a plus “+” sign. If xj = (resp. xj=“+”, xj =“-”) then in the definition of an occurrence of a classical pattern we require ij−ij−1−1 = dj (resp. ij−ij−1−1 ≥ dj , ij−ij−1−1 ≤ dj). For example, if π = 41578362 ∈ S8 then it contains Φ = (132, 31), e.g. π1π5π7 = 486 with distance d = 31, it contains Θ = (312, 03), e.g. π1π2π6 = 413 and π1π2π8 = 412, and it contains Γ = (312, 03−), e.g. π1π2π6 = 413 and π5π6π7 = 836. As a remark, our Definition 3 generalizes the classical and generalized definitions of patterns. For example, avoiding the classical pattern 3421 is the same as avoiding the d-pattern (3421, 000) and avoiding the generalized pattern 3-42-1 is the same as avoiding the pattern (3421, 000). 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+1)n+`((12, d)) = `−1 ∏ j=0 ( (d+1−j)n+`−j n+1 ) d ∏