Resumen de Stable characters from permutation patterns

Christian Gaetz, Christopher Ryba

• For a fixed permutation σ∈Sk, let Nσ denote the function which counts occurrences of σ as a pattern in permutations from Sn. We study the expected value (and dth moments) of Nσ on conjugacy classes of Sn 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,m1,…,mdk which computes these moments on any conjugacy class (of cycle type 1m12m2⋯) 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.