The permutation enumeration of wreath products [formula omitted] of cyclic and symmetric groups

Abstract

Brenti introduced a homomorphism from the symmetric functions to polynomials in one variable with rational coefficients. His map is defined on the elementary symmetric functions. When it is applied to the homogeneous and power symmetric functions the results are generating functions for descents and excedences of permutations in the symmetric group, respectively. Beck and Remmel gave proofs of Brenti's results based on combinatorial definitions of the transition matrices between the bases. In addition, they gave an analog for the hyperoctahedral group. We extend the ideas of these proofs to obtain similar results for wreath products of an arbitrary cyclic group with the symmetric group.