Related papers: On some Euler-Mahonian distributions
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…
Using classical transformations on the symmetric group and two transformations constructed in Fix-Mahonian Calculus I, we show that several multivariable statistics are equidistributed either with the triplet (fix,des,maj), or the pair…
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…
We construct bijections to show that two pairs of sextuple set-valued statistics of permutations are equidistributed on symmetric groups. This extends a recent result of Sokal and the second author valid for integer-valued statistics as…
Recently, we proved the equidistribution of the pairs of permutation statistics $(r\textsf{des},r\textsf{maj})$ and $(r\textsf{exc},r\textsf{den})$. Any pair of permutation statistics that is equidistributed with these pairs is said to be…
Adin, Brenti, and Roichman introduced the pairs of statistics $(\ndes, \nmaj)$ and $(\fdes, \fmaj)$. They showed that these pairs are equidistributed over the hyperoctahedral group $B_n$, and can be considered "Euler-Mahonian" in that they…
The inversion number and the major index are equidistributed on the symmetric group. This is a classical result, first proved by MacMahon, then by Foata by means of a combinatorial bijection. Ever since many refinements have been derived,…
Two well-known distributions in the study of permutation statistics are the Mahonian and Eulerian distributions. Mahonian statistics include the major index MAJ and the number of inversions INV, while examples of Eulerian statistics are the…
We show that the pair (des, ides) of statistics on the set of permu- tations has the same distribution as the pair (asc, row) of statistics on the set of inversion tables, proving a conjecture of Visontai. The common generating function of…
The motivation of this paper is to investigate the joint distribution of succession and Eulerian statistics. We first investigate the enumerators for the joint distribution of descents, big ascents and successions over all permutations in…
A variety of descent and major-index statistics have been defined for symmetric groups, hyperoctahedral groups, and their generalizations. Typically associated to pairs of such statistics is an Euler--Mahonian distribution, a bivariate…
In this paper we look at polynomials arising from statistics on the classes of involutions, $I_n$, and involutions with no fixed points, $J_n$, in the symmetric group. Our results are motivated by F. Brenti's conjecture which states that…
Recently, the second author studied an Eulerian statistic (called cover) in the context of convex polytopes, and proved an equal joint distribution of (cover,des) with (des,exc). In this paper, we present several direct bijective proofs…
We construct two bijections of the symmetric group S_n onto itself that enable us to show that three new three-variable statistics are equidistributed with classical statistics involving the number of fixed points. The first one is…
Our first main result shows that, for words with a fixed multiset of weak right-to-left minima, the statistics within each of the following three classes are equidistributed: 1. Mahonian statistics: $\textsf{inv}$, $\textsf{maj}$,…
The standard algorithm for generating a random permutation gives rise to an obvious permutation statistic $\stat$ that is readily seen to be Mahonian. We give evidence showing that it is not equal to any previously published statistic. Nor…
One of the most central result in combinatorics says that the descent statistic and the excedance statistic are equidistribued over the symmetric group. As a continuation of the work of Shareshian-Wachs (Adv. Math., 225(6) (2010),…
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…
We present (bi-)symmetric generating functions for the joint distributions of Euler-Stirling statistics on permutations, including the number of descents ($\mathsf{des}$), inverse descents ($\mathsf{ides}$), the number of left-to-right…
The Eulerian distribution on the involutions of the symmetric group is unimodal, as shown by Guo and Zeng. In this paper we prove that the Eulerian distribution on the involutions of the hyperoctahedral group, when viewed as a colored…