Symmetric functions, generalized blocks, and permutations with restricted cycle structure

We present various techniques to count proportions of permutations with restricted cycle structure in finite permutation groups. For example, we show how a generalized block theory for symmetric groups, developed by Külshammer, Olsson, and Robinson, can be used for such calculations. The paper includes improvements of recurrence relations of Glasby, results on average numbers of fixed points in certain permutations, and a remark on a conjecture of Robinson related to the so-called k ( G V ) -problem of representation theory. We extend and give alternative proofs for previous results of Erdős and Turán; Glasby; and Beals, Leedham-Green, Niemeyer, Praeger and Seress.