Labeled partitions with colored permutations

In this paper, we extend the notion of labeled partitions with ordinary permutations to colored permutations. We use this structure to derive the generating function of the fmaj k indices of colored permutations. We further give a combinatorial treatment of a relation on the q -derangement numbers with respect to colored permutations. Based on labeled partitions, we provide an involution that implies the generating function formula due to Gessel and Simon for signed q -counting of the major indices. This involution can be extended to signed permutations. This gives a combinatorial interpretation of a formula of Adin, Gessel and Roichman.