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 dth 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 \(n, m_1, \ldots , m_{dk}\) 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 (Adv Appl Math 54:1–10, 2014) and of Gill (The k-assignment polytope, phylogenetic trees, and permutation patterns, Ph.D. Thesis at Linköping University, pp 103–125, 2013), 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.

对于固定排列\(\sigma \in S_k\)，让\(N_{\sigma }\)表示将\(\sigma \) 的出现计数为\(S_n\) 的排列模式的函数。我们研究了\(S_n\)共轭类上\(N_{\sigma }\)的期望值（和第d个矩），并证明这些类函数的不可约字符支持随着n 的增长而稳定。这表示变量\(n, m_1, \ldots , m_{dk}\)中有一个多项式，它计算任何共轭类（循环类型\(1^{m_1}2^{m_2} \cdots \)) 的任何对称群。该结果概括了 Hultman (Adv Appl Math 54:1–10, 2014) 和 Gill (The k-assignment polytope, phylogenetic tree, and permutation patterns, Ph.D. Thesis at Linköping University, pp 103–125, 2013) 的结果)，谁证明了案例\((d,k)=(1,2)\)和 (1, 3) 使用特别的方法。据我们所知，我们的证明是划分代数在排列模式研究中的首次应用。