Related papers: Mahonian STAT on words
We study the expansions of permutation statistics in the basis of functions counting occurrences of a fixed pattern in a permutation. We show the finiteness of these pattern expansions for a class of permutation statistics including the…
Define a permutation to be any sequence of distinct positive integers. Given two permutations p and s on disjoint underlying sets, we denote by p sh s the set of shuffles of p and s (the set of all permutations obtained by interleaving the…
We give a bijective proof of the MacMahon-type equidistribution over the group of signed even permutations C_2 \wr A_n that was stated in [Bernstein. Electron. J. Combin. 11 (2004) 83]. This is done by generalizing the bijection that was…
We give a direct combinatorial proof of the equidistribution of two pairs of permutation statistics, (des, aid) and (lec, inv), which have been previously shown to have the same joint distribution as (exc, maj), the major index and the…
We show that if a permutation statistic can be written as a linear combination of bivincular patterns, then its moments can be expressed as a linear combination of factorials with constant coefficients. This generalizes a result of…
A pair $(\mathrm{st_1}, \mathrm{st_2})$ of permutation statistics is said to be $r$-Euler-Mahonian if $(\mathrm{st_1}, \mathrm{st_2})$ and $( \mathrm{rdes}$, $\mathrm{rmaj})$ are equidistributed over the set $\mathfrak{S}_{n}$ of all…
A partition of the set $[n]:=\{1,2,\ldots,n\}$ is a collection of disjoint nonempty subsets (or blocks) of $[n]$, whose union is $[n]$. In this paper we consider the following rarely used representation for set partitions: given a partition…
We consider the two permutation statistics which count the distinct pairs obtained from the last two terms of occurrences of patterns t_1...t_{m-2}m(m-1) and t_1...t_{m-2}(m-1)m in a permutation, respectively. By a simple involution in…
We study statistics on ordered set partitions whose generating functions are related to $p,q$-Stirling numbers of the second kind. The main purpose of this paper is to provide bijective proofs of all the conjectures of \stein…
We study the distribution and the popularity of some patterns in $k$-ary faro words, i.e. words over the alphabet $\{1, 2, \ldots, k\}$ obtained by interlacing the letters of two nondecreasing words of lengths differing by at most one. We…
We consider a bivariate polynomial that generalizes both the length and reflection length generating functions in a finite Coxeter group. In seeking a combinatorial description of the coefficients, we are led to the study of a new Mahonian…
We prove a conjecture of Haglund which can be seen as an extension of the equidistribution of the inversion number and the major index over permutations to ordered set partitions. Haglund's conjecture implicitly defines two statistics on…
In this paper we provide a bijective proof of a theorem of Garsia and Gessel describing the generating function of the major index over the set of all permutations of [n]={1,...,n} which are shuffles of given disjoint ordered sequences…
Babson and Steingr\'{\i}msson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We consider n-permutations that avoid the generalized pattern…
We obtain new connections between permutation patterns and singularities of Schubert varieties, by giving a new characterization of Gorenstein varieties in terms of so called bivincular patterns. These are generalizations of classical…
Following a result of Eriksen and Sj\"{o}strand (2014) we detail a technique to construct structures following the Fishburn distribution from appropriate Mahonian structures. This technique is introduced on a bivincular pattern of…
Visontai conjectured in 2013 that the joint distribution of ascent and distinct nonzero value numbers on the set of subexcedant sequences is the same as that of descent and inverse descent numbers on the set of permutations. This conjecture…
In the combinatorial study of the coefficients of a bivariate polynomial that generalizes both the length and the reflection length generating functions for finite Coxeter groups, Petersen introduced a new Mahonian statistic $sor$, called…
Various statistics on wreath products are defined via canonical words, "colored" right to left minima and "colored" descents. It is shown that refined counts with respect to these statistics have nice recurrence formulas of…
Recently, Jel\'inek conjectured that there exists a bijection between certain restricted permutations and Fishburn matrices such that the bijection verifies the equidistribution of several statistics. The main objective of this paper is to…