Fix-Euler-Mahonian statistics on wreath products

In 1997 Clarke et al. studied a q-analogue of Eulerʼs difference table for n! using a key bijection Ψ on symmetric groups. In this paper we extend their results to the wreath product of a cyclic group with the symmetric group. In particular we obtain a new Mahonian statistic fmaf on wreath products. We also show that Foata and Hanʼs two recent transformations on the symmetric groups provide indeed a factorization of Ψ.