Recursions for the flag-excedance number in colored permutations groups

The excedance number for S_n is known to have an Eulerian distribution. Nevertheless, the classical proof uses descents rather than excedances. We present a direct proof based on a recursion which uses only excedances and extend it to the flag-excedance parameter defined on the group of colored permutations Gr_,n = ℤ_r ≀ S_n. We have also computed the distribution of a variant of the flag-excedance number, and show that its enumeration uses the Stirling number of the second kind. Moreover, we show that the generating function of the flag-excedance number defined on ℤ_r ≀ S_n is symmetric, and its variant is log-concave on ℤ_r ≀ S_n..