Touchard-Riordan Polynomials and Schur-positivity of Set Partitions

2026/06/11 by Eli Bagno, David Garber · 1 voice
Computer Science · Mathematics · #cs.DM #math.CO

paper · pdf · doi:10.4204/eptcs.445.1

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.

Citations

• Schur-Positivity of Short Chords in Matchings
• Cyclic descents, matchings and Schur-positivity
• Pattern avoidance and quasisymmetric functions
• Block decomposition of permutations and Schur-positivity
• q-Stirling numbers: A new view
• Relations between hypersurface cross ratios, and a combinatorial formula for partitions of a polygon, for permanent preponderance, and for non-associative products
• Motzkin numbers

Discussions

• Eli Bagno, David Garber: Touchard-Riordan Polynomials and Schur-positivity of Set Partitions https://arxiv.org/abs/2606.13149 https://arxiv.org/pdf/2606.13149 https://arxiv.org/html/2606.13149 [bsky, 0 points, 0 comments]

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