Cylindric $P$-Tableaux for (3+1)-Free Posets

TL;DR

This paper introduces cylindric P-tableaux for (3+1)-free posets, defines P-analogs of cylindric Schur functions, and proves their positivity properties, advancing understanding of chromatic symmetric functions and related conjectures.

Contribution

It defines cylindric P-tableaux and P-analogs of cylindric Schur functions, proving their role as generating functions and improving positivity results for chromatic symmetric functions.

Findings

Proved P-analogs of cylindric Schur functions are generating functions of cylindric P-tableaux.

Established positivity of certain sums of e-expansion coefficients of chromatic symmetric functions.

Enhanced evidence for the Stanley-Stembridge conjecture.

Abstract

For a $(3+1)$-free poset $P$, we define a hybrid of $P$-tableaux and cylindric tableaux called cylindric $P$-tableaux. We introduce $P$-analogs of cylindric Schur functions, defined by a determinantal formula, and prove that they are the weight generating functions of cylindric $P$-tableaux. We deduce that certain sums of the $e$-expansion coefficients of the chromatic symmetric function $X_{inc(P)}$ are positive. This improves on Gasharov's theorem on the Schur positivity of $X_{inc(P)}$ and gives further evidence for the Stanley-Stembridge conjecture.