Touchard-Riordan Polynomials and Schur-positivity of Set Partitions
This provides a new combinatorial interpretation and positivity result for symmetric functions, relevant to algebraic combinatorics researchers.
The paper establishes Schur-positivity for symmetric functions associated with set partitions via a descent set function, showing that the expansion coefficients are Touchard-Riordan polynomials counting matchings by crossings.
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.