Some counting problems related topermutation

This paper discusses investigations of sequences of natural numbers which count the orbits of an innnite permutation group on n-sets or n-tuples. It surveys known results on the growth rates, cycle index techniques, and an interpretation as the Hilbert series of a graded algebra, with a possible application to the question of smoothness of growth. I suggest that these orbit-counting sequences are suuciently special to be interesting but suuciently common to support a general theory. `I count a lot of things that there's no need to count,' Cameron said. `Just because that's the way I am. But I count all the things that need to be counted.' Richard Brautigan, The Hawkline Monster 1 Three counting problems This paper is a survey of the problem of counting the orbits of an innnite permutation group on n-sets or n-tuples, especially the aspects closest to algebraic combinatorics. Much of the material surveyed here can be found elsewhere, for example in 4]. We begin by discussing three counting problems in diierent areas of mathematics and their relations. 1.1 Enumeration of nite structures A relational structure M consists of a set X and a family of relations on X. These relations can have arbitrary arities, and there may be a nite or innn-ite number of relations. Many familiar structures have only a single relation: