The relation between burnside rings and combinatorial species

Abstract

We describe the close relationship between permutation groups and combinatorial species (introduced by A. Joyal, Adv. in Math. 42, 1981, 1–82). There is a bijection Φ between the set of transitive actions (up to isomorphism) of S n on finite sets and the set of “molecular” species of degree n (up to isomorphism). This bijection extends to a ring isomorphism between B(S n ) (the Burnside ring of the symmetric group) and the ring V S n (of virtual species of degree n).Since permutation groups are well known (and often studied using computers) this helps in finding examples and counterexamples in species. The cycle index series of a molecular species, which is hard to compute directly, is proved to be simply the (Pólya) cycle polynomial of the corresponding permutation group. Conversely, several operations which are hard to define in Π n B(S n ) have a natural description in terms of species. Both situations are extended to coefficients in λ-rings and binomial rings in the last section.