Touchard-Riordan Polynomials and Schur-positivity of Set Partitions

2026-06-11

AI Summary

This work investigates the Schur positivity of symmetric functions induced by set partitions under a descent-set definition based on adjacent elements within blocks. By integrating tools from symmetric function theory, enumerative combinatorics, and Touchard–Riordan polynomials, the authors establish—for the first time—a direct connection between the Schur expansion coefficients of these symmetric functions over hook-shaped Young diagrams and the Touchard–Riordan polynomials. The results show that these coefficients are precisely given by the Touchard–Riordan polynomials, thereby confirming their nonnegativity and integrality, which establishes Schur positivity. This finding unveils a profound combinatorial link between the enumeration of crossing matchings and the algebraic structure of symmetric functions.

Abstract

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.

Problem

• Schur-positivity
• set partitions
• Touchard-Riordan polynomials
• symmetric functions
• descent set

Innovation

• Schur-positivity
• Touchard-Riordan polynomials
• set partitions
• hook-shaped Young diagrams
• descent set