Cycle type and descent set in wreath products Original Research Article

We express the number of elements of the hyperoctahedral group Bn, which have descent set K and such that their inverses have descent set J, as a scalar product of two representations of Bn. We also give the number of elements of Bn, which have a prescribed descent set and which are in a given conjugacy class of Bn by another scalar product of representations of Bn. 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 Bnʹs.