Reductive enumeration under mutually orthogonal group actions (Q1270635)

This is a substantial paper which describes how, in a fairly general setting, the application of Pólya theory may be aided by considering the effect of a dual action. If a group \(G\) acts on a set \(S\), then it also acts by conjugation on the generating sets of its transitive subgroups, the transitive permutation tuples. If no element of \(G\) acts as the identity, then these actions are orthogonal, that is, every non-trivial element has trivial fixed point set with respect to one of the two actions. This property may substantially reduce the number of terms to be considered in Burnside's theorem, particularly in ``symmetry deficient'' cases. The author applies this technique to count objects in quite a wide variety of situations; e.g, the subgroups of free groups, the number of strong automata, planar maps and plane point confugurations, and coverings of surfaces.