Equidistribution of negative statistics and quotients of Coxeter groups of type B and D

TL;DR

This paper extends known combinatorial identities related to permutation statistics from the symmetric group to Coxeter groups of types B and D, providing new characterizations and generalizations.

Contribution

It introduces generalized identities and characterizations for Coxeter groups of types B and D, expanding classical permutation statistic results.

Findings

Generalized Foata and Schützenberger theorems for types B and D

New characterizations of minimal coset representatives

Extended identities for inversion, major index, and descent distributions

Abstract

We generalize some identities and q-identities previously known for the symmetric group to Coxeter groups of type B and D. The extended results include theorems of Foata and Sch\"utzenberger, Gessel, and Roselle on various distributions of inversion number, major index, and descent number. In order to show our results we provide caracterizations of the systems of minimal coset representatives of Coxeter groups of type B and D.