Enumerating representations in finite wreath products II:: Explicit formulae

We 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 Muller (2000) (the exterior function opi). Building on Theorem I in Muller (2000) and the latter result, we derive explicit formulae for the exponential generating function of the series {\Hom(Gamma, R-n)\} in the case where Gamma is dihedral or a finite abelian group, and the representation sequence {R-n} is any of {H \ S-n} or {H \ A(n)} with a fixed finite group H, or the sequence {W-n} of Weyl groups of type D-n. Moreover, we verify a conjecture concerning the asymptotic behaviour of the sequence {\Hom(G, W-n)\} for finite groups G made in Muller (2000) in the case when G is dihedral or abelian. (C) 2002 Elsevier Science (USA).