  In this paper, we extend the notion of labeled partitions with ordinary
permutations to colored permutations in the sense that the colors are endowed
with a cyclic structure. We use labeled partitions with colored permutations to
derive the generating function of the $\mathrm{fmaj}_k$ indices of colored
permutations. The second result is a combinatorial treatment of a relation on
the q-derangement numbers with respect to colored permutations which leads to
the formula of Chow for signed permutations and the formula of Faliharimalala
and Zeng [10] on colored permutations. The third result is an involution on
permutations that implies the generating function formula for the signed
q-counting of the major indices due to Gessel and Simon. This involution can be
extended to signed permutations. In this way, we obtain a combinatorial
interpretation of a formula of Adin, Gessel and Roichman.
