Related papers: Inverse descent statistic for Andr\'e and simsun p…
Dokos et. al. studied the distribution of two statistics over permutations $\mathfrak{S}_n$ of $\{1,2,\dots, n\}$ that avoid one or more length three patterns. A permutation $\sigma\in\mathfrak{S}_n$ contains a pattern…
We evaluate the probabilities of various events under the uniform distribution on the set of 312-avoiding permutations of 1,...,N. We derive exact formulas for the probability that the ith element of a random permutation is a specific value…
For $\tau\in S_3$, let $S_n(\tau)$ denote the set of permutations in $S_n$ which avoid the pattern $\tau$, and let $E_n^\tau$ denote the expectation with respect to the uniformly random probability measure on $S_n(\tau)$. Let…
The number of inversions is a statistic on permutation groups measuring the degree to which the entries of a permutation are out of order. We provide a generalization of that statistic by introducing the statistic number of pseudoinversions…
We prove several identities expressing polynomials counting permutations by various descent statistics in terms of Eulerian polynomials, extending results of Stembridge, Petersen, and Br\"and\'en. Additionally, we find $q$-exponential…
We investigate permutations in terms of their cycle structure and descent set. To do this, we generalize the classical bijection of Gessel and Reutenauer to deal with permutations that have some ascending and some descending blocks. We then…
We exploit a bijection between plane recursive trees and Stirling permutations; this yields the equivalence of some results previously proven separately by different methods for the two types of objects as well as some new results. We also…
Let $A(n,m)$ denote the Eulerian numbers, which count the number of permutations on $[n]$ with exactly $m$ descents. It is well known that $A(n,m)$ also counts the number of permutations on $[n]$ with exactly $m$ excedances. In this report,…
We introduce inversions tableaux, a new combinatorial model for Schubert polynomials and Stanley symmetric functions that directly specializes to semi-standard Young tableaux in the Grassmannian case. They are a modification of the balanced…
The numbers of even and odd permutations with a given ascent number are investigated using an operator that was previously introduced by the author. Their difference is called a signed Eulerian number. By means of the operator the…
A symmetry of $(t,q)$-Eulerian numbers of type $B$ is combinatorially proved by defining an involution preserving many important statistics on the set of permutation tableaux of type $B$. This involution also proves a symmetry of the…
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…
The study of Mahonian statistics dated back to 1915 when MacMahon showed that the major index and the inverse number have the same distribution on a set of permutations with length n. Since then, many Mahonian statistics have been…
A classical result of MacMahon states that inversion number and major index have the same distribution over permutations of a given multiset. In this work we prove a strengthening of this theorem originally conjectured by Haglund. Our…
The combinatorial theory for the set of parity alternating permutations is expounded. In view of the numbers of ascents and inversions, several enumerative aspects of the set are investigated. In particular, it is shown that signed Eulerian…
The distribution of certain Mahonian statistic (called $\mathrm{BAST}$) introduced by Babson and Steingr\'{i}msson over the set of permutations that avoid vincular pattern $1\underline{32}$, is shown bijectively to match the distribution of…
In this paper we refine the well-known permutation statistic "descent" by fixing parity of (exactly) one of the descent's numbers. We provide explicit formulas for the distribution of these (four) new statistics. We use certain differential…
We prove the equidistribution of several multistatistics over some classes of permutations avoiding a $3$-length pattern. We deduce the equidistribution, on the one hand of inv and foze" statistics, and on the other hand that of maj and…
A permutation statistic $\operatorname{st}$ is said to be shuffle-compatible if the distribution of $\operatorname{st}$ over the set of shuffles of two disjoint permutations $\pi$ and $\sigma$ depends only on $\operatorname{st}\pi$,…
The Eulerian numbers form a triangular array with many interesting properties. The numbers arise from various combinatorial and probabilistic interpretations, and have been studied in a variety of mathematical contexts. In this article we…