Touchard-Riordan Polynomials and Schur-positivity of Set Partitions

A symmetric function is called Schur-positive if it admits an expansion in the Schur basis with nonnegative coefficients.In this paper, we study the Schur-positivity of symmetric functions naturally associated with set partitions, with respect to a descent set function that considers i as descent, if i and i + 1 share a block in the partition.The Schur expansion involves hook-shaped Young diagrams, and the corresponding coefficients are given by Touchard-Riordan polynomials, which enumerate matchings by their number of crossings.1. {i, i + 1} is a matched pair in M.