Some counting problems related to permutation groups

This paper discusses investigations of sequences of natural numbers which count the orbits of an infinite 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 sufficiently special to be interesting but sufficiently common to support a general theory.