Representation theory and cycle statistics for random walks on the symmetric group

We use representation theory of $S_n$ to analyze the mixing of permutation cycle type statistics $a_j(\sigma )=${# of $j$-cycles of $\sigma$} for any fixed $j$ and $\sigma$ resulting from a random $i$-cycle walk on $S_n$. We also derive analogous results for the random star transposition walk. Our approach uses the method of moments; a key ingredient is a new formula for the coefficients in the irreducible character decomposition of the $S_n$-class function (a_j)^r(σ)=\{(\text{# ofj$-cyclesof$σ})^r\} for any positive integers $r,j$ when $n\ge 2rj$.