On fixed points of permutations

The number of fixed points of a random permutation of {1, 2, ..., n} has a limiting Poisson distribution. We seek a generalization, looking at other actions of the symmetric group. Restricting attention to primitive actions, a complete classification of the limiting distributions is given. For most examples, they are trivial - almost every permutation has no fixed points. For the usual action of the symmetric group on k-sets of {1, 2, ..., n}, the limit is a polynomial in independent Poisson variables. This exhausts all cases. We obtain asymptotic estimates in some examples, and give a survey of related results.