Alexander Burstein, Toufik Mansour
We find exact formulas and/or generating functions for the number of words avoiding 3-letter generalized multipermutation patterns and find which of them are equally avoided.
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We find generating functions for the number of words avoiding certain patterns or sets of patterns on at most 2 distinct letters and determine which of them are equally avoided. We also find the exact number of words avoiding certain patterns and provide bijective proofs for the resulting formulas.
We study generating functions for the number of permutations on n letters avoiding 132 and an arbitrary permutation $τ$ on k letters, or containing $τ$ exactly once. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind.
We study generating functions for the number of even (odd) permutations on n letters avoiding 132 and an arbitrary permutation $τ$ on k letters, or containing $τ$ exactly once. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind.
Recently, Babson and Steingrimsson (see [BS]) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We study generating functions for the number of permutations on $n$ letters avoiding $1-3-2$ (or containing $1-3-2$ exactly once) and an arbitrary generalized pattern $τ$ on $k$ letters, or containing $τ$ exactly once. In several cases the generating function depends only on $k$ and is expressed via Chebyshev polynomials of the second kind, and generating function of Motzkin numbers.
It is well known that Universal Cycles of $k$-letter words on an $n$-letter alphabet exist for all $k$ and $n$. In this paper, we prove that Universal Cycles exist for restricted classes of words, including: non-bijections, equitable words (under suitable restrictions), ranked permutations, and "passwords".
A permutation $π$ is said to be {\em Dumont permutations of the first kind} if each even integer in $π$ must be followed by a smaller integer, and each odd integer is either followed by a larger integer or is the last element of $π$ (see, for example, \cite{Z}). In \cite{D} Dumont showed that certain classes of permutations on $n$ letters are counted by the Genocchi numbers. In particular, Dumont showed that the $(n+1)$st Genocchi number is the number of Dummont permutations of the first kind on $2n$ letters. In this paper we study the number of Dumont permutations of the first kind on $n$ let…
A permutation is said to be \emph{alternating} if it starts with rise and then descents and rises come in turn. In this paper we study the generating function for the number of alternating permutations on $n$ letters that avoid or contain exactly once 132 and also avoid or contain exactly once an arbitrary pattern 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 [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…
Recently, Babson and Steingrimsson (see \cite{BS}) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. In this paper we study the generating functions for the number of permutations on $n$ letters avoiding a generalized pattern $ab\mn c$ where $(a,b,c)\in S_3$, and containing a prescribed number of occurrences of generalized pattern $cd\mn e$ where $(c,d,e)\in S_3$. As a consequence, we derive all the previously known results for this kind of problems, as well as many new results.
We define a quantity $c_m(n,k)$ as a generalization of the notion of the composition of the positive integer $n$ into $k$ parts. We proceed to derive some known properties of this quantity. In particular, we relate two partial Bell polynomials, in which the sequence of the variables of one polynomial is the invert transform of the sequence of the variables of the other. We connect the quantities $c_m(n,k)$ and $c_{m-1}(n,k)$ via Pascal matrices. We then relate $c_m(n,k)$ with the numbers of some restricted words over a finite alphabet. We develop a method which transfers some properties of res…
Distribution of sets of descent tops and descent bottoms on restricted permutations
本文研究模式避免排列上下降顶集与下降底集的分布，通过双射证明3字母模式下231、312避免排列的两类集合联合等分布，提出4字母模式相关猜想，且部分猜想已被他人证明。
Define $S_n(R;T)$ to be the number of permutations on $n$ letters which avoid all patterns in the set $R$ and contain each pattern in the multiset $T$ exactly once. In this paper we enumerate $S_n(\{α\};\{β\})$ and $S_n(\emptyset;\{α,β\})$ for all $α\neq β\in S_3$. The results for $S_n(\{α\};\{β\})$ follow from two papers by Mansour and Vainshtein.
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…
Let f_n^r(k) be the number of 132-avoiding permutations on n letters that contain exactly r occurrences of 12... k, and let F_r(x;k) and F(x,y;k) be the generating functions defined by $F_r(x;k)=\sum_{n\gs0} f_n^r(k)x^n$ and $F(x,y;k)=\sum_{r\gs0}F_r(x;k)y^r$. We find an explcit expression for F(x,y;k) in the form of a continued fraction. This allows us to express F_r(x;k) for $1\ls r\ls k$ via Chebyshev polynomials of the second kind.
Let $E_n^r=\{[τ]_a=(τ_1^{(a_1)},...,τ_n^{(a_n)})| τ\in S_n,\ 1\leq a_i\leq r\}$ be the set of all signed permutations on the symbols 1,2,...,n with signs 1,2,...,r. We prove, for every 2-letter signed pattern $[τ]_a$, that the number of $[τ]_a$-avoiding signed permutations in $E_n^r$ is given by the formula $\sum\limits_{j=0}^n j!(r-1)^j{n\choose j}^2$. Also we prove that there are only one Wilf class for r=1, four Wilf classes for r=2, and six Wilf classes for $r\geq 3$.
Pattern avoidance in canon permutations
本文研究任意k下标准排列的经典模式规避，证明规避指定单个模式的计数为k-卡特兰数，枚举相关组合类并建立与k元树的双射，推广相关定理，还利用k正则格词的规避问题枚举特定模式的标准排列并提出猜想。
We complete the enumeration of Dumont permutations of the second kind avoiding a pattern of length 4 which is itself a Dumont permutation of the second kind. We also consider some combinatorial statistics on Dumont permutations avoiding certain patterns of length 3 and 4 and give a natural bijection between 3142-avoiding Dumont permutations of the second kind and noncrossing partitions that uses cycle decomposition, as well as bijections between 132-, 231- and 321-avoiding Dumont permutations and Dyck paths. Finally, we enumerate Dumont permutations of the first kind simultaneously avoiding ce…
Ascent sequences are those consisting of non-negative integers in which the size of each letter is restricted by the number of ascents preceding it and have been shown to be equinumerous with the (2+2)-free posets of the same size. Furthermore, connections to a variety of other combinatorial structures, including set partitions, permutations, and certain integer matrices, have been made. In this paper, we identify all members of the (4,4)-Wilf equivalence class for ascent sequences corresponding to the Catalan number C_n=\frac{1}{n+1}\binom{2n}{n}. This extends recent work concerning avoidance…
Restricted Jacobi permutations
本文系统研究雅可比排列中的模式避免问题，完整枚举了避免长度3模式的雅可比排列，并借助二叉树子族、Dyck路径双射与生成函数技巧，对单一模式限制情形给出了关于上升数、从左到右极小值数及末位字母的精细枚举。
In this paper, we compute and demonstrate the equivalence of the joint distribution of the first letter and descent statistics on six avoidance classes of permutations corresponding to two patterns of length four. This distribution is in turn shown to be equivalent to the distribution on a restricted class of inversion sequences for the statistics that record the last letter and number of distinct positive letters, affirming a recent conjecture of Lin and Kim. Members of each avoidance class of permutations and also of the class of inversion sequences are enumerated by the $n$-th large Schröde…
Tutte founded the theory of enumeration of planar maps in a series of papers in the 1960s. Rooted non-separable planar maps are in bijection with West-2-stack-sortable permutations, beta(1,0)-trees introduced by Cori, Jacquard and Schaeffer in 1997, as well as a family of permutations defined by the avoidance of two four letter patterns. In this paper we give upper and lower bounds on the number of multiple-edge-free rooted non-separable planar maps. We also use the bijection between rooted non-separable planar maps and a certain class of permutations, found by Claesson, Kitaev and Steingrimss…
In this paper, we study the relation between periodicity of two-dimensional words and their abelian pattern complexity. A pattern $\cal{P}$ in $\mathbb{Z}^n$ is the set of all translations of some finite subset $F$ of $\mathbb{Z}^n$. An $F$-factor of an infinite word is a finite word restricted to $F$. Then the pattern complexity over a pattern $\mathcal{P}$ counts the number of distinct $F$-factors of an infinite word, for $P\in \mathcal{P}$. Two finite words are called abelian equivalent if for each letter of the alphabet, they contain the same numbers of occurrences of this letter. The abel…
Goulden and Jackson introduced a very powerful method to study the distributions of certain consecutive patterns in permutations, words, and other combinatorial objects which is now called the cluster method. There are a number of natural classes of combinatorial objects which start with either permutations or words and add additional restrictions. These include up-down permutations, generalized Euler permutations, words with no consecutive repeated letters, Young tableaux, and non-backtracking random walks. We develop an extension of the cluster method which we call the {\em generalized clust…
The monic sequence that shifts left under convolution with itself is the Catalan numbers with 130+ combinatorial interpretations. Here we establish a combinatorial interpretation for the monic sequence that shifts left under composition: it counts permutations that contain a 3241 pattern only as part of a 35241 pattern. We give two recurrences, the first allowing relatively fast computation, the second similar to one for the Catalan numbers. Among the 4 times 4! = 96 similarly restricted patterns involving 4 letters (such as 4\underline{2}31: a 431 pattern only occurs as part of a 4231), four …
In this paper we consider a restricted class of convex polyominoes that we call Z-convex polyominoes. Z-convex polyominoes are polyominoes such that any two pairs of cells can be connected by a monotone path making at most two turns (like the letter Z). In particular they are convex polyominoes, but they appear to resist standard decompositions. We propose a construction by ``inflation'' that allows to write a system of functional equations for their generating functions. The generating function P(t) of Z-convex polyominoes with respect to the semi-perimeter turns out to be algebraic all the s…
We interpret walks in the first quadrant with steps {(1,1),(1,0),(-1,0), (-1,-1)} as a generalization of Dyck words with two sets of letters. Using this language, we give a formal expression for the number of walks in the steps above beginning and ending at the origin. We give an explicit formula for a restricted class of such words using a correspondance between such words and Dyck paths. This explicit formula is exactly the same as that for the degree of the polynomial satisfied by the square of the area of cyclic n-gons conjectured by Dave Robbins although the connection is a mystery. Final…
A partially ordered (generalized) pattern (POP) is a generalized pattern some of whose letters are incomparable, an extension of generalized permutation patterns introduced by Babson and Steingrimsson. POPs were introduced in the symmetric group by Kitaev [Partially ordered generalized patterns, Discrete Math. 298 (2005), 212-229; Introduction to partially ordered patterns, Discrete Appl. Math., to appear], and studied in the set of $k$-ary words by Kitaev and Mansour [Partially ordered generalized patterns and $k$-ary words, Annals of Combinatorics 7 (2003) 191-200]. Moreover, Kitaev et al. […
We consider a Markov chain that iteratively generates a sequence of random finite words in such a way that the $n^{\mathrm{th}}$ word is uniformly distributed over the set of words of length $2n$ in which $n$ letters are $a$ and $n$ letters are $b$: at each step an $a$ and a $b$ are shuffled in uniformly at random among the letters of the current word. We obtain a concrete characterization of the Doob-Martin boundary of this Markov chain. Writing $N(u)$ for the number of letters $a$ (equivalently, $b$) in the finite word $u$, we show that a sequence $(u_n)_{n \in \mathbb{N}}$ of finite words c…
Counting $r\times s$ rectangles in nondecreasing and Smirnov words
本文研究了非降字与 Smirnov 字中 $r\ imes s$ 矩形容量的分布，求得了其二元生成函数及总数生成函数，补充了关于一般字和 Catalan 字的近期研究。
A Smirnov word is a word over the positive integers in which adjacent letters must be different. A symmetric function enumerating these words by descent number arose in the work of Shareshian and the second named author on $q$-Eulerian polynomials, where a $t$-analog of a formula of Carlitz, Scoville, and Vaughan for enumerating Smirnov words is proved. A symmetric function enumerating a circular version of these words by cyclic descent number arose in the work of the first named author on chromatic quasisymmetric functions of directed graphs, where a $t$-analog of a formula of Stanley for enu…
A poset is said to be (2+2)-free if it does not contain an induced subposet that is isomorphic to 2+2, the union of two disjoint 2-element chains. Two elements in a poset are indistinguishable if they have the same strict up-set and the same strict down-set. Being indistinguishable defines an equivalence relation on the elements of the poset. We introduce the statistic maxindist, the maximum size of a set of indistinguishable elements. We show that, under a bijection of Bousquet-Melou et al., indistinguishable elements correspond to letters that belong to the same run in the so-called ascent s…
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