Euler-Mahonian triple set-valued statistics on permutations

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 number of descents des and excedances exc. Also of interest are pairs of permutation statistics that have the same joint distribution as (des, MAJ); these are called Euler Mahonian pairs. In this paper we define a Stirling statistic as one...

We prove that the pair of statistics (des,maj) on multiset permutations is equidistributed with the pair (stc,inv) on certain quotients of the symmetric group. We define the analogue of the statistic stc on multiset permutations, whose joint distribution with the inversions equals that of (des,maj). We extend the definition of the statistic stc to hyperoctahedral and even hyperoctahedral groups. Such functions, together with the Coxeter length, are equidistributed, respective...

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 number of excedances of a permutation. Moreover, the triple (pix, lec, inv) was shown to have the same distribution as (fix, exc, maj), where fix is the number of fixed points of a permutation. We define a new statistic aix so that our bijectio...

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 permutations of $\{1,2,\ldots, n\}$, where $\mathrm{rdes}$ denotes the $r$-descent number and $\mathrm{rmaj}$ denotes the $r$-major index introduced by Rawlings. The main objective of this paper is to prove that $(\mathrm{exc}_r, \mathrm{den}_r...

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 ordered set partitions and states that they are equidistributed. The implied inversion statistic is equivalent to a statistic on ordered set partitions studied by Steingr\'imsson, Ishikawa, Kasraoui, and Zeng, and is known to have a nice dist...

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 does its joint distribution with the standard Eulerian statistics $\des$ and $\exc$ appear to coincide with any known Euler-Mahonian pair. A general construction of Skandera yields an Eulerian partner $\ska$ such that $(\ska, \stat)$ is e...

We propose a unified approach to prove general formulas for the joint distribution of an Eulerian and a Mahonian statistic over a set of colored permutations by specializing Poirier's colored quasisymmetric functions. We apply this method to derive formulas for Euler-Mahonian distributions on colored permutations, derangements and involutions. A number of known formulas are recovered as special cases of our results, including formulas of Biagioli-Zeng, Assaf, Haglund-Loehr-Re...

Recently Petersen defined a new Mahonian index sor over the symmetric group $\mathfrak{S}_n$ and proved that $(\text{inv}, \text{rlmin})$ and $(\text{sor}, \text{cyc})$ have the same joint distribution. Foata and Han proved that the pairs of set-valued statistics $(\text{Cyc}, \text{Rmil}), (\text{Cyc}, \text{Lmap}), (\text{Rmil}, \text{Lmap})$ have the same joint distribution over $\mathfrak{S}_n$. In this paper we introduce the set-valued statistics $\text{Inv}, \text{Lmi...

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 makl statistics, over these classes of pattern avoiding permutations. Here inv and maj are the celebrated Mahonian statistics, foze" is one of the statistics defined in terms of generalized patterns in the 2000 pioneering paper of Babson and S...

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 of $[n]$ with blocks $B_{1},B_{2},\ldots,B_{m}$ satisfying $\max B_{1}<\max B_{2}<\cdots<\max B_{m}$, we represent it by a word $w=w_{1}w_{2}\ldots w_{n}$ such that $i\in B_{w_{i}}$, $1\leq i\leq n$. We prove that the Mahonian statistics I...