Schur-Positivity of Short Chords in Matchings

TL;DR

This paper proves the Schur-positivity of certain matchings with fixed unmatched vertices, providing two proofs, deriving coefficients, and exploring various refined Schur-positive sets and their combinatorial properties.

Contribution

It introduces a new combinatorial criterion and a bijective proof for Schur-positivity in matchings, along with characterizations and interpretations of coefficients.

Findings

Matchings with fixed unmatched vertices are Schur-positive.

Coefficients in the Schur expansion relate to Bessel polynomials.

Characterization of matchings whose avoidance sets are Schur-positive.

Abstract

We prove that the set of matchings with a fixed number of unmatched vertices is Schur-positive with respect to the set of short chords. Two proofs are presented. The first proof applies a new combinatorial criterion for Schur-positivity, while the second is bijective. The coefficients in the Schur expansion are derived, and interpreted in terms of Bessel polynomials. We present a Knuth-like equivalence relation on matchings, and show that every equivalence class corresponds to an irreducible representation. We proceed to find various refined Schur-positive sets, including the set of matchings with a prescribed crossing number and the set of matchings with a given number of pairs of intersecting chords. Finally, we characterize all the matchings $m$ such that the set of matchings avoiding $m$ is Schur-positive.