Quadrant marked mesh patterns in 132-avoiding permutations III

Given a permutation σ = σ1 . . . σn in the symmetric group Sn, we say that σi matches the marked mesh pattern MMP(a, b, c, d) in σ if there are at least a points to the right of σi in σ which are greater than σi, at least b points to the left of σi in σ which are greater than σi, at least c points to the left of σi in σ which are smaller than σi, and at least d points to the right of σi in σ which are smaller than σi. This paper is continuation of the systematic study of the distributions of quad- rant marked mesh patterns in 132-avoiding permutations started in [9] and [10] where we studied the distribution of the number of matches of MMP(a, b, c, d) in 132-avoiding permutations where at most two elements of a, b, c, d are greater than zero and the remaining elements are zero. In this paper, we study the distribution of the number of matches of MMP(a, b, c, d) in 132-avoiding permutations where at least three of a, b, c, d are greater than zero. We provide explicit recurrence relations to enumerate our objects which can be used to give closed forms for the generating functions associated with such distributions. In many cases, we provide combinatorial explanations of the coefficients that appear in our generating functions

Combinatorics of simple marked mesh patterns in 132-avoiding permutations

We present some combinatorial interpretations for coefficients appearing in series partitioning the permutations avoiding 132 along marked mesh patterns. We identify for patterns in which only one parameter is non zero the combinatorial family in bijection with 132-avoiding permutations and also preserving the statistic counted by the marked mesh pattern.Comment: 11 pages, 6 figures, submitted at FPSAC 201

Quadrant marked mesh patterns in 123-avoiding permutations

Given a permutation $\sigma = \sigma_1 \ldots \sigma_n$ in the symmetric group $\mathcal{S}_{n}$, we say that $\sigma_i$ matches the quadrant marked mesh pattern $\mathrm{MMP}(a,b,c,d)$ in $\sigma$ if there are at least $a$ points to the right of $\sigma_i$ in $\sigma$ which are greater than $\sigma_i$, at least $b$ points to the left of $\sigma_i$ in $\sigma$ which are greater than $\sigma_i$, at least $c$ points to the left of $\sigma_i$ in $\sigma$ which are smaller than $\sigma_i$, and at least $d$ points to the right of $\sigma_i$ in $\sigma$ which are smaller than $\sigma_i$. Kitaev, Remmel, and Tiefenbruck systematically studied the distribution of the number of matches of $\mathrm{MMP}(a,b,c,d)$ in 132-avoiding permutations. The operation of reverse and complement on permutations allow one to translate their results to find the distribution of the number of $\mathrm{MMP}(a,b,c,d)$ matches in 231-avoiding, 213-avoiding, and 312-avoiding permutations. In this paper, we study the distribution of the number of matches of $\mathrm{MMP}(a,b,c,d)$ in 123-avoiding permutations. We provide explicit recurrence relations to enumerate our objects which can be used to give closed forms for the generating functions associated with such distributions. In many cases, we provide combinatorial explanations of the coefficients that appear in our generating functions

Harmonic numbers, Catalan's triangle and mesh patterns

The notion of a mesh pattern was introduced recently, but it has already proved to be a useful tool for description purposes related to sets of permutations. In this paper we study eight mesh patterns of small lengths. In particular, we link avoidance of one of the patterns to the harmonic numbers, while for three other patterns we show their distributions on 132-avoiding permutations are given by the Catalan triangle. Also, we show that two specific mesh patterns are Wilf-equivalent. As a byproduct of our studies, we define a new set of sequences counted by the Catalan numbers and provide a relation on the Catalan triangle that seems to be new

Simple marked mesh patterns

In this paper we begin the first systematic study of distributions of simple marked mesh patterns. Mesh patterns were introduced recently by Br\"and\'en and Claesson in connection with permutation statistics. We provide explicit generating functions in several general cases, and develop recursions to compute the numbers in question in some other cases. Certain $q$-analogues are discussed. Moreover, we consider two modifications of the notion of a marked mesh pattern and provide enumerative results for them.Comment: 27 page

The 1-box pattern on pattern avoiding permutations

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

On the combinatorics of quadrant marked mesh patterns in 132-avoiding permutations

International audienceThe study of quadrant marked mesh patterns in 132-avoiding permutations was initiated by Kitaev, Remmel and Tiefenbruck. We refine several results of Kitaev, Remmel and Tiefenbruck by giving new combinatorial interpretations for the coefficients that appear in the generating functions for the distribution of quadrant marked mesh patterns in 132-avoiding permutations. In particular , we study quadrant marked mesh patterns where we specify conditions on exactly one of the four possible quadrants in a quadrant marked mesh pattern. We show that for the first two quadrants, certain of these coefficients are counted by elements of Catalan's triangle and give a new combinatorial interpretation of these coefficients for quadrant four. We also give a new bijection between 132-avoiding permutations and non-decreasing parking functions

Singleton mesh patterns in multidimensional permutations

This paper introduces the notion of mesh patterns in multidimensional permutations and initiates a systematic study of singleton mesh patterns (SMPs), which are multidimensional mesh patterns of length 1. A pattern is avoidable if there exist arbitrarily large permutations that do not contain it. As our main result, we give a complete characterization of avoidable SMPs using an invariant of a pattern that we call its rank. We show that determining avoidability for a $d$-dimensional SMP $P$ of cardinality $k$ is an $O(d\cdot k)$ problem, while determining rank of $P$ is an NP-complete problem. Additionally, using the notion of a minus-antipodal pattern, we characterize SMPs which occur at most once in any $d$-dimensional permutation. Lastly, we provide a number of enumerative results regarding the distributions of certain general projective, plus-antipodal, minus-antipodal and hyperplane SMPs.Comment: Theorem 12 and Conjecture 1 are replaced by a more general Theorem 12; the paper is to appear in JCT

Quadrant marked mesh patterns in alternating permutations

This paper is continuation of the systematic study of distribution of quadrant marked mesh patterns. We study quadrant marked mesh patterns on up-down and down-up permutations, also known as alternating and reverse alternating permutations, respectively. In particular, we refine classic enumeration results of Andr\'{e} on alternating permutations by showing that the distribution of the quadrant marked mesh pattern of interest is given by $(\sec(xt))^{1/x}$ on up-down permutations of even length and by $\int_0^t (\sec(xz))^{1+\frac{1}{x}}dz$ on down-up permutations of odd length.Comment: 20 page

Quadrant marked mesh patterns in 132-avoiding permutations I

This paper is a continuation of the systematic study of the distributions of quadrant marked mesh patterns initiated in [6]. Given a permutation \sg = \sg_1 ... \sg_n in the symmetric group $S_n$, we say that \sg_i matches the quadrant marked mesh pattern $MMP(a,b,c,d)$ if there are at least $a$ elements to the right of \sg_i in \sg that are greater than \sg_i, at least $b$ elements to left of \sg_i in \sg that are greater than \sg_i, at least $c$ elements to left of \sg_i in \sg that are less than \sg_i, and at least $d$ elements to the right of \sg_i in \sg that are less than \sg_i. We study the distribution of $MMP(a,b,c,d)$ in 132-avoiding permutations. In particular, we study the distribution of $MMP(a,b,c,d)$, where only one of the parameters $a,b,c,d$ are non-zero. In a subsequent paper [7], we will study the the distribution of $MMP(a,b,c,d)$ in 132-avoiding permutations where at least two of the parameters $a,b,c,d$ are non-zero.Comment: Theorem 10 is correcte