Toufik Mansour
We study the generating function for the number of involutions on $n$ letters containing exactly $r\gs0$ occurrences of 3412. It is shown that finding this function for a given $r$ amounts to a routine check of all involutions on $2r+1$ letters.
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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。
A double occurrence word $w$ over a finite alphabet $Σ$ is a word in which each alphabet letter appears exactly twice. Such words arise naturally in the study of topology, graph theory, and combinatorics. Recently, double occurrence words have been used for studying DNA recombination events. We develop formulas for counting and enumerating several elementary classes of double occurrence words such as palindromic, irreducible, and strongly-irreducible words.
We introduce Peano words, which are words corresponding to finite approximations of the Peano space filling curve. We then find the number of occurrences of certain patterns in these words.
We consider sigma-words, which are words used by Evdokimov in the construction of the sigma-sequence. We then find the number of occurrences of certain patterns and subwords in these words.
We find generating functions the number of strings (words) containing a specified number of occurrences of certain types of order-isomorphic classes of substrings called subword patterns. In particular, we find generating functions for the number of strings containing a specified number of occurrences of a given 3-letter subword pattern.
We study the generating function for the number of even (or odd) permutations on n letters containing exactly $r\gs0$ occurrences of 132. It is shown that finding this function for a given r amounts to a routine check of all permutations in $S_{2r}$.
We establish a combinatorial connection between the sequence $(i_{n,k})$ counting the involutions on $n$ letters with $k$ descents and the sequence $(a_{n,k})$ enumerating the semistandard Young tableaux on $n$ cells with $k$ symbols. This allows us to show that the sequences $(i_{n,k})$ are not log-concave for some values of $n$, hence answering a conjecture due to F. Brenti.
Involution words are variations of reduced words for involutions in Coxeter groups, first studied under the name of "admissible sequences" by Richardson and Springer. They are maximal chains in Richardson and Springer's weak order on involutions. This article is the first in a series of papers on involution words, and focuses on their enumerative properties. We define involution analogues of several objects associated to permutations, including Rothe diagrams, the essential set, Schubert polynomials, and Stanley symmetric functions. These definitions have geometric interpretations for certain …
We count the number of occurrences of certain patterns in given words. We choose these words to be the set of all finite approximations of a sequence generated by a morphism with certain restrictions. The patterns in our considerations are either classical patterns 1-2, 2-1, 1-1-...-1, or arbitrary generalized patterns without internal dashes, in which repetitions of letters are allowed. In particular, we find the number of occurrences of the patterns 1-2, 2-1, 12, 21, 123 and 1-1-...-1 in the words obtained by iterations of the morphism 1->123, 2->13, 3->2, which is a classical example of a m…
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 a notion of pattern occurrence that generalizes both classical permutation patterns as well as poset containment. Many questions about pattern statistics and avoidance generalize naturally to this setting, and we focus on functional complexity problems -- particularly those that arise by constraining the order dimensions of the pattern and text posets. We show that counting the number of induced, injective occurrences among dimension 2 posets is #P-hard; enumerating the linear extensions that occur in realizers of dimension 2 posets can be done in polynomial time, while for uncons…
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…
The problem of counting plane trees with $n$ edges and an even or an odd number of leaves was studied by Eu, Liu and Yeh, in connection with an identity on coloring nets due to Stanley. This identity was also obtained by Bonin, Shapiro and Simion in their study of Schröder paths, and it was recently derived by Coker using the Lagrange inversion formula. An equivalent problem for partitions was independently studied by Klazar. We present three parity reversing involutions, one for unlabelled plane trees, the other for labelled plane trees and one for 2-Motzkin paths which are in one-to-one corr…
Babson and Steingr\'ımsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Claesson presented a complete solution for the number of permutations avoiding any single pattern of type $(1,2)$ or $(2,1)$. For eight of these twelve patterns the answer is given by the Bell numbers. For the remaining four the answer is given by the Catalan numbers. With respect to being equidistributed there are three different classes of patterns of type $(1,2)$ or $(2,1)$. We present a recursion for the number of perm…
Counting occurrences of a pattern in a binary word
本文提出字典序极值引用方法，给出二元词中出现次数的计算公式，并构造算法枚举包含给定词 p 的二元词及本原词，以研究 B_{n,p}(k)。
We apply ideas from the cluster method to q-count the permutations of a multiset according to the number of occurrences of certain generalized patterns, as defined by Babson and Steingrimsson. In particular, we consider those patterns with three letters and one internal dash, as well as permutation statistics composed of counting the number of occurrences of multisets of such patterns. Counting is done via recurrences which simplify in the case of permutations. A collection of Maple procedures implementing these recurrences accompanies the article.
The word inference problem is to determine languages such that the information on the number of occurrences of those subwords in the language can uniquely identify a word. A considerable amount of work has been done on this problem, but the same cannot be said for circular words despite growing interests on the latter due to their applications -- for example, in splicing systems. Meanwhile, Parikh matrices are useful tools and well established in the study of subword occurrences. In this work, we propose two ways of counting subword occurrences in circular words. We then extend the idea of Par…
We count the number of distinct (scattered) subwords occurring in the base-b expansion of the non-negative integers. More precisely, we consider the sequence $(S_b(n))_{n\ge 0}$ counting the number of positive entries on each row of a generalization of the Pascal triangle to binomial coefficients of base-$b$ expansions. By using a convenient tree structure, we provide recurrence relations for $(S_b(n))_{n\ge 0}$ leading to the $b$-regularity of the latter sequence. Then we deduce the asymptotics of the summatory function of the sequence $(S_b(n))_{n\ge 0}$.
Counting occurrences of patterns in permutations
本文提出一种多重集合元素计数新方法，并将其用于研究排列中模式出现次数，扩展长度3模式结果，系统分析长度4模式并猜想七个Wilf类的渐近行为。
We prove that the generating function for the number of flattened permutations having a given number of occurrences of the pattern 13-2 is rational, by using the recurrence relations and the kernel method.
For a given language $L$, we study the languages $X$ such that for all distinct words $u, v \in L$, there exists a word $x \in X$ that appears a different number of times as a factor in $u$ and in $v$. In particular, we are interested in the following question: For which languages $L$ does there exist a finite language $X$ satisfying the above condition? We answer this question for all regular languages and for all sets of factors of infinite words.
The Goulden-Jackson cluster method is a powerful tool for counting words by occurrences of prescribed subwords, and was adapted by Elizalde and Noy for counting permutations by occurrences of prescribed consecutive patterns. In this paper, we lift the cluster method for permutations to the Malvenuto-Reutenauer algebra. Upon applying standard homomorphisms, our result specializes to both the cluster method for permutations as well as a q-analogue which keeps track of the inversion number statistic. We construct additional homomorphisms using the theory of shuffle-compatibility, leading to furth…
In their study of cyclic pattern containment, Domagalski et al. conjecture differential equations for the generating functions of circular permutations avoiding consecutive patterns of length 3. In this note, we prove and significantly generalize these conjectures. We show that, for every consecutive pattern $σ$ beginning with 1, the bivariate generating function counting occurrences of $σ$ in circular permutations can be obtained from the generating function counting occurrences of $σ$ in (linear) permutations. This includes all the patterns for which the latter generating function is known.
In this paper, we begin with the Lehman-Walsh formula counting one-face maps and construct two involutions on pairs of permutations to obtain a new formula for the number $A(n,g)$ of one-face maps of genus $g$. Our new formula is in the form of a convolution of the Stirling numbers of the first kind which immediately implies a formula for the generating function $A_n(x)=\sum_{g\geq 0}A(n,g)x^{n+1-2g}$ other than the well-known Harer-Zagier formula. By reformulating our expression for $A_n(x)$ in terms of the backward shift operator $E: f(x)\rightarrow f(x-1)$ and proving a property satisfied b…
In this paper, we extend the notion of labeled partitions with ordinary permutations to colored permutations in the sense that the colors are endowed with a cyclic structure. We use labeled partitions with colored permutations to derive the generating function of the $\mathrm{fmaj}_k$ indices of colored permutations. The second result is a combinatorial treatment of a relation on the q-derangement numbers with respect to colored permutations which leads to the formula of Chow for signed permutations and the formula of Faliharimalala and Zeng [10] on colored permutations. The third result is an…
Given a set $Π$ of permutation patterns of length at most $k$, we present an algorithm for building $S_{\le n}(Π)$, the set of permutations of length at most $n$ avoiding the patterns in $Π$, in time $O(|S_{\le n - 1}(Π)| \cdot k + |S_{n}(Π)|)$. Additionally, we present an $O(n!k)$-time algorithm for counting the number of copies of patterns from $Π$ in each permutation in $S_n$. Surprisingly, when $|Π| = 1$, this runtime can be improved to $O(n!)$, spending only constant time per permutation. Whereas the previous best algorithms, based on generate-and-check, take exponential time per permutat…
Recently, Kitaev and Remmel [Classifying descents according to parity, Annals of Combinatorics, to appear 2007] refined the well-known permutation statistic ``descent'' by fixing parity of one of the descent's numbers. Results in that paper were extended and generalized in several ways. In this paper, we shall fix a set partition of the natural numbers $N$, $(N_1, ..., N_t)$, and we study the distribution of descents, levels, and rises according to whether the first letter of the descent, rise, or level lies in $N_i$ over the set of words over the alphabet $[k]$. In particular, we refine and g…
Counting permutations with no long monotone subsequence via generating trees and the kernel method
本文通过生成树和核方法，重新证明并推广了Gessel关于无长单调子序列排列的生成函数行列式公式，并扩展到对合情形，同时跟踪固定点数。
Counting $r\times s$ rectangles in (Catalan) words
本文提出字的矩形容量统计量以计数条形图中的 $r\ imes s$ 矩形，并推导普通字与 Catalan 字的分布及总数生成函数。
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。
An avoidance pattern where the letters within an occurrence of which are required to be adjacent is referred to as a subword. In this paper, we enumerate members of the set NC_n of non-crossing partitions of length n according to the number of occurrences of several infinite families of subword patterns each containing repeated letters. As a consequence of our results, we obtain explicit generating function formulas counting the members of NC_n for n >= 0 according to all subword patterns of length three containing a repeated letter. Further, simple expressions are deduced for the total number…
The Goulden-Jackson cluster method is a powerful tool for obtaining generating functions for counting words in a free monoid by occurrences of a set of subwords. We introduce a generalization of the cluster method for monoid networks, which generalize the combinatorial framework of free monoids. As a sample application of the generalized cluster method, we compute bivariate and multivariate generating functions counting Motzkin paths---both with height bounded and unbounded---by statistics corresponding to the number of occurrences of various subwords, yielding both closed-form and continued f…
A perfect matching M in an edge-colored complete bipartite graph K_{n,n} is rainbow if no pair of edges in M have the same color. We obtain asymptotic enumeration results for the number of rainbow matchings in terms of the maximum number of occurrences of a color. We also consider two natural models of random edge-colored K_{n,n} and show that, if the number of colors is at least n, then there is with high probability a random matching. This in particular shows that almost every square matrix of order n in which every entry appears at most n times has a Latin transversal.
A redundant generating function is a generating function having terms which are not part of the solution of the original problem. We use redundant generating functions to study two path problems. In the first application we explain a surprising occurrence of Catalan numbers in counting paths that stay below the line y = 2x. In the second application we prove a conjecture of Niederhausen and Sullivan.
Counting subgraphs of coloring graphs
本文将色多项式推广为计数着色图中任意导出子图的$π_G^{(H)}(k)$，证明其关于颜色数$k$为多项式，研究了色对多项式并揭示了其与树度序列的等价关系，同时生成了新的图不变量。
We study relationships between permutation statistics and pattern-functions, counting the number of times particular patterns occur in a permutation. This allows us to write several familiar statistics as linear combinations of pattern counts, both in terms of a permutation and in terms of its image under the fundamental bijection. We use these enumerations to resolve the question of characterizing so-called "shallow" permutations, whose depth (equivalently, disarray/displacement) is minimal with respect to length and reflection length. We present this characterization in several ways, includi…
We prove several general formulas for the distributions of various permutation statistics over any set of permutations whose quasisymmetric generating function is a symmetric function. Our formulas involve certain kinds of plethystic substitutions on quasisymmetric generating functions, and the permutation statistics we consider include the descent number, peak number, left peak number, and the number of up-down runs. We apply these results to cyclic permutations, involutions, and derangements, and more generally, to derive formulas for counting all permutations by the above statistics jointly…
On the Partial Sum of Subword-Counting Sequences
本文研究了二进制展开中子词出现次数的部分和 $\sum_{n=0}^N(-1)^{s_w(n)}$ 的行为，并刻画了满足该部分和为 $O(N^{1-ε})$ 的几类词 $w$。
Generating functions for fixed points of the Mullineux map
该研究将Andrews-Olsson与Bessenrodt-Olsson关于Mullineux映射不动点计数的已有结果推广到任意e的情形，确定了对应的生成函数，补充了偶数e下的新结果。
In this short note we look at the problem of counting juggling patterns with one ball or two balls with a throw at every occurrence. We will do this for both traditional juggling and for spherical juggling. In the latter case we will show a connection to the "associated Mersenne numbers" (A001350) and so as a result will be able to recover a proof that the $p$th Lucas number is congruent to 1 modulo p when p is a prime.
A group action on cyclic compositions and $γ$-positivity
本文通过平面树的内部节点刻画及循环复合上的群作用，给出 Dyck 路径计数的新解释，并组合证明 Narayana 多项式与相关 W 多项式的 $\gamma$-正性。
The Ramsey number $R(P_1,P_2,...,P_t;r)$ is a valve value such that as long as the cardinality $n$ of the $n$-set $V_n={1,...,n}$ is no less than $R$,however all the $\binom{n}{r}$ $r$-subsets of $V_n$ are distributed into $t$ boxes, $V_n$ will always have a property $W$ expressed as eq.(1).Thus, by calculating the number of ways of distribution of $r$-subsets that makes $W$ true,one can get an equation for $R(P_1,P_2,...,P_t;r)$.The evaluation of the general term in this eq. and the counting of the frequencies of occurrence of the various values the general term takes can be reduced to the pr…
In a previous paper, we studied an overpartition analogue of Gaussian polynomials as the generating function for overpartitions fitting inside an $m \times n$ rectangle. Here, we add one more parameter counting the number of overlined parts, obtaining a two-parameter generalization $\overline{m+n \brack n}_{q,t}$ of Gaussian polynomials, which is also a $(q,t)$-analogue of Delannoy numbers. First we obtain finite versions of classical $q$-series identities such as the $q$-binomial theorem and the Lebesgue identity, as well as two-variable generalizations of classical identities involving Gauss…
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