### U.W. Bangor - School of Informatics - Mathematics Preprints 1991

# Computational discrete algebra

#### 91.15 : MORRIS, I. & WENSLEY, C.D.

### 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.

#### 91.16 : MORRIS, I. & WENSLEY, C.D.

### 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.