# Computational discrete algebra

### Cycle indices and subgroup lattices

#### Abstract:

We determine the properties of the conjugate \check{\tau} = \phi^{-1} \Delta \phi of a diagonal function \Delta by the mark function \phi in the incidence algebra of the poset of conjugacy classes of subgroups of a finite group G. Particular choices for \Delta provide applications to the Burnside ring of G, to the theory of \beta-rings and to Polya-Redfield enumeration. In particular, we obtain a Polya-like substitution formula for the K[J]-inventory of colourings of a set whose symmetry group is the wreath product G[F].

#### Published in:

Discrete Mathematics 118 (1993) 173-195.

### lambda-Operations in beta-Rings

#### Abstract:

A \beta-ring is supplied with operations \beta_H where H runs over the conjugacy classes of subgroups of the symmetric groups S_n. In an earlier paper we introduced a second set of operations \lambda_H and we show here that the two sets are related by the isomorphism \beta_H(-X) \cong (-1)^n \lambda_H(X).
We then consider the operations \beta_H and \lambda_H as combinatorial species, in the sense of Joyal, and express their molecular decomposition as a finite sum of products of the exponential species with molecular species of degree at most n. We give combinatorial interpretations for \beta_{S_n}-structures and \lambda_{S_n}-structures and derive various species isomorphisms.

#### Published in:

Math. Proc. Camb. Phil. Soc. 121 (1997) 247-267.