Hook coefficients of chromatic functions

TL;DR

This paper investigates the positivity of Schur function coefficients, specifically hook shapes, in the chromatic symmetric function of graphs, and establishes a correspondence with P-tableaux for certain poset incomparability graphs.

Contribution

It advances understanding of Schur positivity in chromatic symmetric functions by focusing on hook shapes and relates graph orientations to P-tableaux for special posets.

Findings

Proves positivity of Schur coefficients for hook shapes.

Establishes a correspondence between acyclic orientations and P-tableaux.

Provides new insights into Stanley's and Gasharov's positivity conjectures.

Abstract

The chromatic symmetric function of a graph is a generalization of the chromatic polynomial. The key motivation for studying the structure of a chromatic symmetric function is to answer positivity conjectures by Stanley in 1995 and Gasharov in 1996. As a symmetric function one can write the chromatic symmetric function in the basis of Schur functions. In this paper we address the positivity of the Schur coefficients when the parameter of the Schur function is a hook shape. Furthermore, when a graph is the incomparability graph of a poset with specific properties we construct a correspondence between acyclic orientations of the graph and P-tableaux with a hook shape.