Stable characters from permutation patterns

For a fixed permutation $\sigma \in S_k$, let $N_{\sigma}$ denote the function which counts occurrences of $\sigma$ as pattern in permutations from $S_n$. We study expected value (and $d$-th moments) on conjugacy classes $S_n$ and prove that irreducible character support these class functions stabilizes $n$ grows. This says there is single polynomial variables $n, m_1, \ldots, m_{dk}$ computes moments any (of cycle type $1^{m_1}2^{m_2}\cdots$) symmetric group. result generalizes results Hultman Gill, who proved cases $(d,k)=(1,2)$ $(1,3)$ using ad hoc methods. Our proof is, to our knowledge, first application partition algebras patterns.