Signed Euler-Mahonian identities

TL;DR

This paper generalizes signed Euler-Mahonian identities from symmetric groups to Coxeter groups of types B, D, and complex reflection groups, providing new identities and sign-balance polynomials.

Contribution

It extends signed Euler-Mahonian identities to broader Coxeter groups using generalized bijections, introducing new identities and sign-balance polynomials.

Findings

Extended identities to types B, D, and G(r,1,n)

Derived new sign-balance polynomials for types B and D

Provided bijective proofs for the generalized identities

Abstract

A relationship between signed Eulerian polynomials and the classical Eulerian polynomials on $\mathfrak{S}_n$ was given by D\'{e}sarm\'{e}nien and Foata in 1992, and a refined version, called signed Euler-Mahonian identity, together with a bijective proof were proposed by Wachs in the same year. By generalizing this bijection, in this paper we extend the above results to the Coxeter groups of types $B_n$, $D_n$, and the complex reflection group $G(r,1,n)$, where the `sign' is taken to be any one-dimensional character. Some obtained identities can be further restricted on some particular set of permutations. We also derive some new interesting sign-balance polynomials for types $B_n$ and $D_n$.