Distance Enumerators for Permutation Groups

Abstract

We consider the distance enumerator Δ _G (x) of a finite permutation group G, which is the polynomial ∑ _g∈G x ^n−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 Δ _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 et al. (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 C _m  ≀ S _n .

Key Words:

• Cycle index
• Distance enumerator
• Permutation group

Mathematics Subject Classification:

• Primary 20B05
• Secondary 05A15, 94B60

Notes

Communicated by H. D. Macpherson.