Positivity of permutation pattern character polynomials

Abstract

Let Nσ(π) denote the number of occurrences of a permutation pattern σ ∈ S k in a permutation π ∈ Sn. Gaetz and Ryba [3] showed using partition algebras that the d-th moment M σ,d,n (π) of Nσ on the conjugacy class of π is given by a polynomial in n, m1, . . . , m dk , where mi denotes the number of i-cycles of π. They also show that the coefficient χ λ [n] , M σ,d,n agrees with a polynomial a λ σ,d (n) in n. This work is motivated by our conjecture that when σ = id k is the identity permutation, all of these coefficients a λ id k (n) are nonnegative. We directly compute closed forms for these polynomials in the cases λ = (1), (1, 1), and (2), and use this to verify our positivity conjecture for those cases by showing that the polynomials are real-rooted with all roots less than k. We also study the case a(1) σ (n), for which we give a formula for the polynomials and their leading coefficients.