Enumerating Representations in Finite Wreath Products II: Explicit Formulae

AbstractWe present several contributions to the enumerative theory of wreath product representations developed in a previous paper by the first named author (Adv. in Math.153 (2000), 118–154). Theorem 3.1 of the present paper establishes an explicit formula for one of the key ingredients in the description of the corresponding generating functions given in Müller (2000) (the exterior function ΦΓ). Building on Theorem 1 in Müller (2000) and the latter result, we derive explicit formulae for the exponential generating function of the series {∣Hom (Γ,Rn)∣} in the case where Γ is dihedral or a finite abelian group, and the representation sequence {Rn} is any of {H≀Sn} or {H≀An} with a fixed finite group H, or the sequence {∣Hom(G, Wn)}. Moreover, we verify a conjecture concerning the asymptotic behaviour of the sequence \{∣Hom(G,Wn)∣\} for finite groups G made in Müller (2000) in the case when G is dihedral or abelian