Euler-Mahonian Statistics via Polyhedral Geometry

TL;DR

This paper employs polyhedral geometry to derive new multivariate generalizations of Euler--Mahonian distributions for permutation statistics, providing a unified framework and bijective proofs of their equivalences.

Contribution

It introduces a novel polyhedral geometric approach to generalize and unify known Euler--Mahonian distributions for various permutation groups.

Findings

Established new multivariate Euler--Mahonian identities

Provided bijective proofs of distribution equivalences

Unified various known distribution formulas

Abstract

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 generating function identity encoding these statistics. We use techniques from polyhedral geometry to establish new multivariate generalizations for many of the known Euler--Mahonian distributions. The original bivariate distributions are then straightforward specializations of these multivariate identities. A consequence of these new techniques are bijective proofs of the equivalence of the bivariate distributions for various pairs of statistics.