Cycle type and descent set in wreath products

We express the number of elements of the hyperoctahedral group B_n, which have descent set K and such that their inverses have descent set J, as a scalar product of two representations of B_n. We also give the number of elements of B_n, which have a prescribed descent set and which are in a given conjugacy class of B_n by another scalar product of representations of B_n. For this, we first establish corresponding results for certain wreath products. We finally give, by generating series of symmetric functions, some analogs of the classical formulas which express the exponential generating series of alternating elements in the B_n's.