Permutation statistics of products of random permutations

Abstract

Given a permutation statistic s:Sn→R, define the mean statistics¯ as the class function giving the mean of s over conjugacy classes. We describe a way to calculate the expected value of s on a product of t independently chosen elements from the uniform distribution on a union of conjugacy classes Γ⊆Sn. In order to apply the formula, one needs to express the class function s¯ as a linear combination of irreducible Sn-characters. We provide such expressions for several commonly studied permutation statistics, including the exceedance number, inversion number, descent number, major index and k-cycle number. In particular, this leads to formulae for the expected values of said statistics.