Positivity of permutation pattern character polynomials

Abstract

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