Fix-euler-mahonian Statistics on Wreath Products

In 1997 Clarke, Han, and Zeng introduced a q-analogue of Euler’s 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. By generalizing their bijection Ψ we prove the equidistribution of the triple statistics (fix, exc, fmaj) and (fix, exc, fmaf) on wreath products, where “fix”, “exc”, “fmaj” and “fmaf” denote the number of fixed points, the number of excedances, the flag major index and the flag maf index, respectively. As a consequence we obtain a new mahonian statistic fmaf on wreath products. Finally we show that Foata and Han’s two recent transformations on the symmetric groups provide indeed a factorization of Ψ.