Signed Euler–Mahonian identities

Abstract A relationship between signed Eulerian polynomials and the classical Eulerian polynomials on S n was given by Desarmenien and Foata in 1992, and a refined version, called signed Euler–Mahonian identity, together with a bijective proof was 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 .