Eric S. Egge, Toufik Mansour
We use combinatorial and generating function techniques to enumerate various sets of involutions which avoid 231 or contain 231 exactly once. Interestingly, many of these enumerations can be given in terms of $k$-generalized Fibonacci numbers.
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The Bordelaise philosophy, or rather a juvenile version of it, is used to enumerate self avoiding walks in a $[0,1] \times (- \infty, \infty)$.
We study the involutions belonging to the class of 321 avoiding permutations. We calculate the algebraic generating functions of the set containing the involutions avoiding 321 and of some of its subsets. Precisely we determine the algebraic generating functions of the involutions that are expansions of 12, of those expansions of 21, of the simple ones and of their expansions. The graphics of the simple involutions are caracterized. Being the simple involutions avoiding 321 counted by Riordan's numbers, a combinatoric interpretation of the results is illustrated through a class of Motzkin path…
We investigate permutations and involutions that avoid a pattern of length three and have a {\em unique} longest increasing subsequence.
We consider the enumeration of pattern-avoiding involutions, focusing in particular on sets defined by avoiding a single pattern of length 4. As we demonstrate, the numerical data for these problems demonstrates some surprising behavior. This strange behavior even provides some very unexpected data related to the number of 1324-avoiding permutations.
Riordan arrays, denoted by pairs of generating functions (g(z), f(z)), are infinite lower-triangular matrices that are used as combinatorial tools. In this paper, we present Riordan and stochastic Riordan arrays that have connections to the Fibonacci and modified Lucas numbers. Then, we present some pseudo-involutions in the Riordan group that are based on constructions starting with a certain generating function g(z). We also present a theorem that shows how to construct pseudo-involutions in the Riordan group starting with a certain generating function f(z) whose additive inverse has composi…
We complete the Wilf classification of signed patterns of length 5 for both signed permutations and signed involutions. New general equivalences of patterns are given which prove Jaggard's conjectures concerning involutions in the symmetric group avoiding certain patterns of length 5 and 6. In this way, we also complete the Wilf classification of S_5, S_6, and S_7 for both permutations and involutions.
In this note, we study the mean length of the longest increasing subsequence of a uniformly sampled involution that avoids the pattern $3412$ and another pattern.
Odd-indexed Fibonacci numbers via pattern-avoiding permutations
本文研究了若干组合问题，其计数结果为奇数索引斐波那契数，包括特定类型的Dyck路径、块喷泉、有向列凸多边形和无交叉无嵌套的集合划分。通过构造到避免模式排列的双射，并推导生成函数以跟踪排列层面的统计量。
In 1990 West conjectured that there are $2(3n)!/((n+1)!(2n+1)!)$ two-stack sortable permutations on $n$ letters. This conjecture was proved analytically by Zeilberger in 1992. Later, Dulucq, Gire, and Guibert gave a combinatorial proof of this conjecture. In the present paper we study generating functions for the number of two-stack sortable permutations on $n$ letters avoiding (or containing exactly once) 132 and avoiding (or containing exactly once) an arbitrary permutation $τ$ on $k$ letters. In several interesting cases this generating function can be expressed in terms of the generating f…
We study the fine structure of the sets of involutions avoiding either 4312 (I(4321)) or 3412 (I(3412)), connecting the point of view of the decomposition theorems with the one of the associated labelled Motzkin paths. The algebraic generating function of the simple involutions in I(4321) is given, together with other generating functions, while the set I(3412) is shown containing no simple involutions of length n>2. The reverse-complement bijection maintains the fine structures of I(4321) and trivially of I(3412).
We enumerate and characterize some classes of alternating and reverse alternating involutions avoiding a single pattern of length three or four. If on one hand the case of patterns of length three is trivial, on the other hand, the length four case is more challenging and involves sequences of combinatorial interest, such as Motzkin and Fibonacci numbers.
Centrosymmetric involutions in the symmetric group S_{2n} are permutations πsuch that π=π^{-1} and π(i)+π(2n+1-i)=2n+1 for all i, and they are in bijection with involutions of the hyperoctahedral group. We describe the distribution of some natural descent statistics on 321-avoiding centrosymmetric involutions, including the number of descents in the first half of the involution, and the sum of the positions of these descents. Our results are based on two new bijections, one between centrosymmetric involutions in S_{2n} and subsets of {1,...,n}, and another one showing that certain statistics o…
A centrosymmetric permutation is one which is invariant under the reverse-complement operation, or equivalently one whose associated standard Young tableaux under the Robinson-Schensted algorithm are both invariant under the Schutzenberger involution. In this paper, we characterize the set of permutations avoiding 1243 and 2143 whose images under the reverse-complement mapping also avoid these patterns. We also characterize in a simple manner the corresponding Schroder paths under a bijection of Egge and Mansour. We then use these results to enumerate centrosymmetric permutations avoiding the …
We show that the distribution of the major index over the set of involutions in S_n that avoid the pattern 321 is given by the q-analogue of the n-th central binomial coefficient. The proof consists of a composition of three non-trivial bijections, one being the Robinson-Schensted correspondence, ultimately mapping those involutions with major index m into partitions of m whose Young diagram fits inside an n/2 by n/2 box. We also obtain a refinement that keeps track of the descent set, and we deduce an analogous result for the comajor index of 123-avoiding involutions.
We use a sign-reversing involution to show that trees on the vertex set [n], considered to be rooted at 1, in which no vertex has exactly one child are counted by 1/n sum_{k=1}^{n} (-1)^(n-k) {n}-choose-{k} (n-1)!/(k-1)! k^(k-1). This result corrects a persistent misprint in the Encyclopedia of Integer Sequences.
Involutions avoiding 4321 and another pattern of length four
本文通过Biane双射和生成函数，枚举了避免4321及另一个长度四模式的对合族，给出了精确的二元生成函数和结构描述，并对剩余情况提供了函数方程和连分数表示。
In [GM] Guibert and Mansour studied involutions on n letters avoiding (or containing exactly once) 132 and avoiding (or containing exactly once) an arbitrary pattern on k letters. They also established a bijection between 132-avoiding involutions and Dyck word prefixes of same length. Extending this bijection to bilateral words allows to determine more parameters; in particular, we consider the number of inversions and rises of the involutions onto the words. This is the starting point for considering two different directions: even/odd involutions and statistics of some generalized patterns. T…
We prove that $|Av_n(231,312,1432)|$, $|Av_n(312,321,1342)|$ $|Av_n(231,312,4321,21543)|$, and $ |Av_n(321,231,4123,21534)|$, are all equal to $F_{n+1} - 1$ where $F_n$ is the $n$-th Fibonacci number using the convention $F_0 = F_1 = 1$ and $Av_n(S)$ is the set of all permutations of length $n$ that avoid all of the patterns in the set $S$. To do this, we characterize the structures of the permutations in these sets in terms of Fibonacci permutations. Then, we further quantify the structures using statistics such as inversion number and a statistic that measures the length of Fibonacci subsequ…
Several authors have examined connections among restricted permutations, continued fractions, and Chebyshev polynomials of the second kind. In this paper we prove analogues of these results for involutions which avoid 3412. Our results include a recursive procedure for computing the generating function for involutions which avoid 3412 and any set of additional patterns. We use our results to give enumerations and generating functions for involutions which avoid 3412 and various sets of additional patterns. In many cases we express these generating functions in terms of Chebyshev polynomials of…
We consider the set partition statistics ls and rb introduced by Wachs and White and investigate their distribution over set partitions avoiding certain patterns. In particular, we consider those set partitions avoiding the pattern 13/2, $Π_n(13/2)$, and those avoiding both 13/2 and 123, $Π_n(13/2,123)$. We show that the distribution over $Π_n(13/2)$ enumerates certain integer partitions, and the distribution over $Π_n(13/2,123)$ gives q-Fibonacci numbers. These q-Fibonacci numbers are closely related to q-Fibonacci numbers studied by Carlitz and by Cigler. We provide combinatorial proofs that…
We study generating functions for the number of involutions in $S_n$ avoiding (or containing once) 132, and avoiding (or containing once) an arbitrary permutation $τ$ on $k$ letters. In several interesting cases the generating function depends only on $k$ and is expressed via Chebyshev polynomials of the second kind. In particular, we establish that involutions avoiding both 132 and $12... k$ have the same enumerative formula according to the length than involutions avoiding both 132 and any {\em double-wedge pattern} possibly followed by fixed points of total length $k$. Many results are also…
Wythoff-Fibonacci Sequences and a Perturbed Greedy Almost-involution
引入下、上 Wythoff-Fibonacci 序列，证明它们划分自然数集，并用于给出序列 q*_j 的显式公式，该序列是自然数集的置换且满足 q*_{q*_j}=j（j≥5），即近对合。
Enumerating Pattern-Avoiding Involutions using Combinatorial Exploration
针对避免模式的对合枚举研究较少的问题，推导两类4-模式避免对合的Wilf等价类的代数生成函数，调整Mosaic方法扩展其余两类序列初始项并分析其渐近行为。
This thesis deals with three different aspects of the combinatorics of permutations. In the first two papers, two flavours of pattern avoiding permutations are examined; and in the third paper Young tableaux, which are closely related to permutations via representation theory, are studied. In the first paper we give solutations to several interesting problems regarding pattern avoiding doubly alternating permutations, such as finding a bijection between 1234-avoiding permutations and 1234-avoiding doubly alternating permutations of twice the size. In the second paper partial permutations which…
We provide a trace partitioned Gray code for the set of q-ary strings avoiding a pattern constituted by k consecutive equal symbols. The definition of this Gray code is based on two different constructions, according to the parity of q. This result generalizes, and is based on, a Gray code for binary strings avoiding k consecutive 0's.
We show how a bijection due to Biane between involutions and labelled Motzkin paths yields bijections between Motzkin paths and two families of restricted involutions that are counted by Motzkin numbers, namely, involutions avoiding 4321 and 3412. As a consequence, we derive characterizations of Motzkin paths corresponding to involutions avoiding either 4321 or 3412 together with any pattern of length 3. Furthermore, we exploit the described bijection to study some notable subsets of the set of restricted involutions, namely, fixed point free and centrosymmetric restricted involutions.
In this paper we prove a refined major-balance identity on the $321$-avoiding involutions of length $n$, respecting the leading element of permutations. The proof is based on a sign-reversing involution on the lattice paths within a $\lfloor\frac{n}{2}\rfloor\times\lceil\frac{n}{2}\rceil$ rectangle. Moreover, we prove affirmatively a question about refined major-balance identity on the $123$-avoiding involutions, respecting the number of descents.
There are (at least) three bijections from Dyck paths to 321-avoiding permutations in the literature, due to Billey-Jockusch-Stanley, Krattenthaler, and Mansour-Deng-Du. How different are they? Denoting them B,K,M respectively, we show that M = B \circ L = K \circ L' where L is the classical Kreweras-Lalanne involution on Dyck paths and L', also an involution, is a sort of derivative of L. Thus K^{-1} \circ B, a measure of the difference between B and K, is the product of involutions L' \circ L and turns out to be a very curious bijection: as a permutation on Dyck n-paths it is an nth root of …
Let $A_k$ be the set of permutations in the symmetric group $S_k$ with prefix 12. This paper concerns the enumeration of involutions which avoid the set of patterns $A_k$. We present a bijection between symmetric Schroder paths of length $2n$ and involutions of length $n+1$ avoiding $\mathcal{A}_4$. Statistics such as the number of right-to-left maxima and fixed points of the involution correspond to the number of steps in the symmetric Schroder path of a particular type. For each $k> 2$ we determine the generating function for the number of involutions avoiding the subsequences in $A_k$, acco…
Asymptotic distribution of fixed points of pattern-avoiding involutions
该论文研究模式避免对合中固定点数量的极限分布，涵盖任意长度单调模式及所有长度3模式，利用杨表联系与生成函数方法，揭示了多种已知分布。
Limit Theorems for Fixed Point Biased Pattern Avoiding Involutions
本文研究了避免特定模式的固定点偏向置换，通过分析不同模式的偏置强度，揭示了固定点数量的渐近分布特性。
We examine the enumeration of certain Motzkin objects according to the numbers of crossings and nestings. With respect to continued fractions, we compute and express the distributions of the statistics of the numbers of crossings and nestings over three sets, namely the set of $4321$-avoiding involutions, the set of $3412$-avoiding involutions, and the set of $(321,3\bar{1}42)$-avoiding permutations. To get our results, we exploit the bijection of Biane restricted to the sets of $4321$- and $3412$-avoiding involutions which was characterized by Barnabei et al.~ and the bijection between $(321,…
It is well known that permutations avoiding any 3-length pattern are enumerated by the Catalan numbers. If the three patterns 123, 132 and 213 are avoided at the same time we obtain a class of permutations enumerated by the Fibonacci numbers. We start from these permutations and make one or two forbidden patterns disappear by suitably "generalizing" them. In such a way we find several classes of permutations enumerated by integer sequences which lay between the Fibonacci and Catalan numbers. For each class, we provide the generating function according to the length of the permutations. Moreove…
We give multiple proofs of two formulas concerning the enumeration of permutations avoiding a monotone consecutive pattern with a certain value for the inverse peak number or inverse left peak number statistic. The enumeration in both cases is given by a sequence related to Fibonacci numbers. We also show that there is exactly one permutation whose inverse peak number is zero among all permutations with any fixed descent composition, and we give a few elementary consequences of this fact. Our proofs involve generating functions, symmetric functions, regular expressions, and monomino-domino til…
This paper is continuation of the study of the 1-box pattern in permutations introduced by the authors in \cite{kitrem4}. We derive a two-variable generating function for the distribution of this pattern on 132-avoiding permutations, and then study some of its coefficients providing a link to the Fibonacci numbers. We also find the number of separable permutations with two and three occurrences of the 1-box pattern.
Let B_n be the hyperoctahedral group; that is, the set of all signed permutations on n letters, and let B_n(T) be the set of all signed permutations in B_n which avoids a set T of signed patterns. In this paper, we find all the cardinalities of the sets B_n(T) where $T \subseteq B_2$. This allow us to express these cardinalities via inverse of binomial coefficients, binomial coefficients, Catalan numbers, and Fibonacci numbers.
We count permutations avoiding a nonconsecutive instance of a two- or three-letter pattern, that is, the pattern may occur but only as consecutive entries in the permutation. Two-letter patterns give rise to the Fibonacci numbers. The counting sequences for the two representative three-letter patterns, 321 and 132, have respective generating functions (1+x^2)(C(x)-1)/(1+x+x^2-x C(x)) and C(x+x^3) where C(x) is the generating function for the Catalan numbers.
Let $\mathcal I_n$ and $\mathcal J_n$ denote the set of involutions and fixed-point free involutions of $\{1, \dots, n\}$, respectively, and let $\text{des}(π)$ denote the number of descents of the permutation $π$. We prove a conjecture of Guo and Zeng which states that $I_n(t) := \sum_{π\in \mathcal I_n} t^{\text{des}(π)}$ is $γ$-positive for $n \ge 1$ and $J_{2n}(t) := \sum_{π\in \mathcal J_{2n}} t^{\text{des}(π)}$ is $γ$-positive for $n \ge 9$. We also prove that the number of $(3412, 3421)$-avoiding permutations with $m$ double descents and $k$ descents is equal to the number of separable …
The past decade has seen a flurry of research into pattern avoiding permutations but little of it is concerned with their exhaustive generation. Many applications call for exhaustive generation of permutations subject to various constraints or imposing a particular generating order. In this paper we present generating algorithms and combinatorial Gray codes for several families of pattern avoiding permutations. Among the families under consideration are those counted by Catalan, Schröder, Pell, even index Fibonacci numbers and the central binomial coefficients. Consequently, this provides Gray…
We study $321$-avoiding affine permutations, and prove a formula for their enumeration with respect to the inversion number by using a combinatorial approach. This is done in two different ways, both related to Viennot's theory of heaps. First, we encode these permutations using certain heaps of monomers and dimers. This method specializes to the case of affine involutions. For the second proof, we introduce periodic parallelogram polyominoes, which are new combinatorial objects of independent interest. We enumerate them by extending the approach of Bousquet-Mélou and Viennot used for classica…
The number of even 321-avoiding permutations of length n is equal to the number of odd ones if n is even, and exceeds it by the (n-1)/2th Catalan number otherwise. We present an involution that proves a refinement of this sign-balance property respecting the length of the longest increasing subsequence of the permutation. In addition, this yields a combinatorial proof of a recent analogous result of Adin and Roichman dealing with the last descent. In particular, we answer the question how to obtain the sign of a 321-avoiding permutation from the pair of tableaux resulting from the Robinson-Sch…
In 2012, Sagan and Savage introduced the notion of $st$-Wilf equivalence for a statistic $st$ and for sets of permutations that avoid particular permutation patterns which can be extended to generalized permutation patterns. In this paper we consider $inv$-Wilf equivalence on sets of two or more consecutive permutation patterns. We say that two sets of generalized permutation patterns $Π$ and $Π'$ are $inv$-Wilf equivalent if the generating function for the inversion statistic on the permutations that simultaneously avoid all elements of $Π$ is equal to the generating function for the inversio…
A \Def{composition} of a positive integer $n$ is a $k$-tuple $(ł_1, ł_2, \dots, ł_k) \in \Z_{> 0}^k$ such that $n = ł_1 + ł_2 + \dots + ł_k$. Our goal is to enumerate those compositions whose parts $ł_1, ł_2, \dots, ł_k$ avoid a fixed arithmetic sequence. When this sequence is given by the even integers (i.e., all parts of the compositions must be odd), it is well known that the number of compositions is given by the Fibonacci sequence. A much more recent theorem says that when the parts are required to avoid all multiples of a given integer $k$, the resulting compositions are counted by a seq…
A permutation $π$ contains a pattern $σ$ if and only if there is a subsequence in $π$ with its letters are in the same relative order as those in $σ$. Partially ordered patterns (POPs) provide a convenient way to denote patterns in which the relative order of some of the letters does not matter. This paper elucidates connections between the avoidance sets of a few POPs with other combinatorial objects, directly answering five open questions posed by Gao and Kitaev \cite{gao-kitaev-2019}. This was done by thoroughly analysing the avoidance sets and developing recursive algorithms to derive thes…
In a recent paper, Backelin, West and Xin describe a map $ϕ^*$ that recursively replaces all occurrences of the pattern $k... 21$ in a permutation $σ$ by occurrences of the pattern $(k-1)... 21 k$. The resulting permutation $ϕ^*(σ)$ contains no decreasing subsequence of length $k$. We prove that, rather unexpectedly, the map $ϕ^*$ commutes with taking the inverse of a permutation. In the BWX paper, the definition of $ϕ^*$ is actually extended to full rook placements on a Ferrers board (the permutations correspond to square boards), and the construction of the map $ϕ^*$ is the key step in provi…
We introduce the $k$-bonacci polyominoes, a new family of polyominoes associated with the binary words avoiding $k$ consecutive $1$'s, also called generalized $k$-bonacci words. The polyominoes are very entrancing objects, considered in combinatorics and computer science. The study of polyominoes generates a rich source of combinatorial ideas. In this paper we study some properties of $k$-bonacci polyominoes. Specifically, we determine their recursive structure and, using this structure, we enumerate them according to their area, semiperimeter, and length of the corresponding words. We also in…
Lehmer Codes and the Reverse-Complement Mapping from (32-1)-Avoiding Permutations to (3-21)-Avoiding Permutations
该研究以避免特定vincular模式的排列为对象，借助莱默码将其转化为加权偏序集，证明其极大元由斐波那契数计数，且经典逆补映射在极大元间构成双射，揭示了偏序集结构的对称性。
We study the distribution of the statistics 'number of fixed points' and 'number of excedances' in permutations avoiding subsets of patterns of length 3. We solve all the cases of simultaneous avoidance of more than one pattern, giving generating functions enumerating these two statistics. Some cases are generalized to patterns of arbitrary length. For avoidance of one single pattern we give partial results. We also describe the distribution of these statistics in involutions avoiding any subset of patterns of length 3. The main technique is to use bijections between pattern-avoiding permutati…
Counting permutations avoiding two flat partially ordered patterns
研究同时避免两个平坦部分有序模式的排列计数问题，通过建立与k-斐波那契数的联系和双射方法，推导生成函数并扩展了单个平坦POPs的计数结果。
A simple permutation is one that does not map a nontrivial interval onto an interval. It was recently proved by Albert and Atkinson that a permutation class with only finitely simple permutations has an algebraic generating function. We extend this result to enumerate permutations in such a class satisfying additional properties, e.g., the even permutations, the involutions, the permutations avoiding generalised permutations, and so on.
We show that two classes of combinatorial objects--inversion tables with no subsequence of decreasing consecutive numbers and matchings with no 2-nestings--are enumerated by the Fishburn numbers. In particular, we give a simple bijection between matchings with no 2-nestings and inversion tables with no subsequence of decreasing consecutive numbers. We then prove using the involution principle that inversion tables with no subsequence of decreasing consecutive numbers have the same generating function as the Fishburn numbers. The Fishburn numbers have previously been shown by Bousquet-Mélou, Cl…
The study of pattern avoidance in permutations, and specifically in flattened partitions is an active area of current research. In this paper, we count the number of distinct flattened partitions over [n] avoiding a single pattern, as well as a pair of two patterns. Several counting sequences, namely Catalan numbers, powers of two, Fibonacci numbers and Motzkin numbers arise. We also consider other combinatorial statistics, namely runs and inversions, and establish some bijections in situations where the statistics coincide.
A permutation is simsun if for all k, the subword of the one-line notation consisting of the k smallest entries does not have three consecutive decreasing elements. Simsun permutations were introduced by Simion and Sundaram, who showed that they are counted by the Euler numbers. In this paper we enumerate simsun permutations avoiding a pattern or a set of patterns of length 3. The results involve Motkzin, Fibonacci, and secondary structure numbers. The techniques in the proofs include generating functions, bijections into lattice paths and generating trees.
Random Fibonacci Words via Clone Schur Functions
本文通过克隆Schur函数生成随机斐波那契词，研究其正性、概率性质及与多项式、正交序列等的联系，扩展了Plancherel测度的渐进行为。
Despite the fact that the field of pattern avoiding permutations has been skyrocketing over the last two decades, there are very few exhaustive generating algorithms for such classes of permutations. In this paper we introduce the notions of regular and colored regular set of forbidden patterns, which are particular cases of right-justified sets of forbidden patterns. We show the (colored) regularity of several sets of forbidden patterns (some of them involving variable length patterns) and we derive a general framework for the efficient generation of permutations avoiding them. The obtained g…
Inversion sequences are integer sequences $e=e_{1}e_{2}\dots e_{n}$ such that $0\leq e_{i}<i$ for each $i$. The study of patterns in inversion sequences was initiated by Corteel--Martinez--Savage--Weselcouch and Mansour--Shattuck in the classical (non-consecutive) case, and later by Auli--Elizalde in the consecutive case, where the entries of a pattern are required to occur in adjacent positions. In this paper we continue this investigation by considering {\em consecutive patterns of relations}, in analogy to the work of Martinez--Savage in the classical case. Specifically, given two binary re…
In this paper we define and study what we call the double Catalan monoid. This monoid is the image of a natural map from the 0-Hecke monoid to the monoid of binary relations. We show that the double Catalan monoid provides an algebraization of the (combinatorial) set of 4321-avoiding permutations and relate its combinatorics to various off-shoots of both the combinatorics of Catalan numbers and the combinatorics of permutations. In particular, we give an algebraic interpretation of the first derivative of the Kreweras involution on Dyck paths, of 4321-avoiding involutions and of recent results…
A Fishburn permutation is a permutation which avoids the bivincular pattern $(231, \{1\}, \{1\})$, while an ascent sequence is a sequence of nonnegative integers in which each entry is less than or equal to one more than the number of ascents to its left. Fishburn permutations and ascent sequences are linked by a bijection $g$ of Bousquet-Mélou, Claesson, Dukes, and Kitaev. We write $F_n(σ_1,\ldots,σ_k)$ to denote the set of Fishburn permutations of length $n$ which avoid each of $σ_1,\ldots,σ_k$ and we write $A_n(α_1,\ldots,α_k)$ to denote the set of ascent sequences which avoid each of $α_1,…
The distribution of Coxeter descents and block number over the set of fully commutative elements in the hyperoctahedral group $B_n$, $\FC(B_n)$, is studied in this paper. We prove that the associated Chow quasi-symmetric generating function is equal to a non-negative sum of products of two Schur functions. The proof involves a decomposition of $\FC(B_n)$ into a disjoint union of two-sided Barbash-Vogan combinatorial cells, a type $B$ extension of Rubey's descent preserving involution on $321$-avoiding permutations and a detailed study of the intersection of $\FC(B_n)$ with $S_n$-cosets which y…
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