On some Euler-Mahonian distributions

TL;DR

This paper establishes new equidistribution results for permutation statistics, extending classical distributions to multiset permutations and hyperoctahedral groups, and introduces analogues of the stc statistic.

Contribution

It introduces the analogue of the stc statistic for multiset permutations and extends these distributions to hyperoctahedral groups, broadening the scope of Euler-Mahonian distributions.

Findings

(des,maj) is equidistributed with (stc,inv) on certain quotients

The stc analogue on multiset permutations matches (des,maj) distribution

Extended distributions to hyperoctahedral and even hyperoctahedral groups

Abstract

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, respectively, with (ndes,nmaj) and (ddes,dmaj).