Fix-Euler-Mahonian statistics on wreath products

In 1997 Clarke et al. studied a $q$-analogue of Euler’s difference table for $n!$ using a key bijection $\mathrm{\Psi }$ 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 \emph{fmaf} on wreath products. We also show that Foata and Han’s two recent transformations on the symmetric groups provide indeed a factorization of $\mathrm{\Psi }$.