Stable characters from permutation patterns

Abstract

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

11 pages

Cited by in corpus (3)

• On pattern avoidance in matchings and involutions
• Moments of permutation statistics and central limit theorems
• Positivity of permutation pattern character polynomials