Distance enumerators for permutation groups

We consider the distance enumerator ∆G(x) of a finite permutation group G, which is the polynomial ∑ g∈G xn−π(g), where n is the degree of G and π(g) the number of fixed points of g ∈ G. In particular, we introduce a bivariate polynomial which is a special case of the cycle index of G, and from which ∆G(x) can be obtained, and then use this new polynomial to prove some identities relating the distance enumerators of groups G and H with those of their direct and wreath products. In the case of the direct product, this answers a question of Blake, Cohen and Deza from 1979. We also use the identity for the wreath product to find an explicit combinatorial expression for the distance enumerators of the generalised hyperoctahedral groups Cm oSn.