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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 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.

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We are grateful to Axel Hultman for his help with important references and to Pavel Etingof for his helpful comments.

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Correspondence to Christian Gaetz.

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C.G. was supported by a National Science Foundation Graduate Research Fellowship under Grant No. 1122374.

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Gaetz, C., Ryba, C. Stable characters from permutation patterns. Sel. Math. New Ser. 27, 70 (2021).

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