Signed Euler-Mahonian identities
Sen-Peng Eu, Zhicong Lin, Yuan-Hsun Lo

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 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 , , and the complex reflection group , 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 and .
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