Fixed points of non-uniform permutations and representation theory of the symmetric group

Abstract

We use representation theory of the symmetric group S_n to prove Poisson limit theorems for the distribution of fixed points for three types of non-uniform permutations. First, we give results for the commutator of g and x where g and x are uniform in S_n. Second, we give results for the commutator of g and x where g in uniform in S_n and x is fixed. Third, we give results for permutations obtained by multiplying n*log(n)/i + cn many random i-cycles. Some of our results are known by other, quite different, methods.

16 pages.This is a major rewriting. Thanks to Jimmy He a conjecture from the previous draft is proved. We also incorporate feedback from Alex Miller