Signed Permutation Statistics

We derive multivariate generating functions that count signed permutations by various statistics, using the hyperoactahedral generalization of methods of Garsia and Gessel. We also derive the distributions over inverse descent classes of signed permutations for two of these statistics individually (the major index and inversion number). These results show that, in contrast to the case for (unsigned) permutations, these two statistics are not generally equidistributed. We also discuss applications to statistics on the wreath product Ck ≀ Sn of a cyclic group with the symmetric group.