Signed Euler–Mahonian identities

A relationship between signed Eulerian polynomials and the classical on Sn was given by Désarménien Foata in 1992, a refined version, called Euler–Mahonian identity, together with bijective proof proposed Wachs same year. By generalizing this bijection, paper we extend above results to Coxeter groups of types Bn, Dn, complex reflection group G(r,1,n), where ‘sign’ is taken be any one-dimensional character. Some obtained identities can further restricted some particular set permutations. We also derive new interesting sign-balance for Bn Dn.