Planar graphs and Stanley's Chromatic Functions

TL;DR

This paper explores positivity phenomena in chromatic symmetric functions of graphs, providing new combinatorial proofs and interpretations, especially for planar graphs and certain classes of posets, advancing understanding of their algebraic and combinatorial properties.

Contribution

It introduces a new combinatorial interpretation of Schur-coefficients via planar networks and offers simplified proofs of positivity results for chromatic functions of specific graph classes.

Findings

A new combinatorial proof of Gasharov's Schur-positivity theorem.

A combinatorial interpretation of Schur-coefficients using planar networks.

Reproves monomial positivity of G-analogues of power sum symmetric functions.

Abstract

This article is dedicated to the study of positivity phenomena for the chromatic symmetric function of a graph with respect to various bases of symmetric functions. We give a new proof of Gasharov's theorem on the Schur-positivity of the chromatic symmetric function of a $(3 + 1)$-free poset. We present a combinatorial interpretation of the Schur-coefficients in terms of planar networks. Compared to Gasharov's proof, it gives a clearer visual illustration of the cancellation procedures and is quite similar in spirit to the proof of monomial positivity of Schur functions via the Lindstrom-Gessel-Viennot lemma. We apply a similar device to the $e$-positivity problem of chromatic functions. Following Stanley, we analyze certain analogs of symmetric functions attached to graphs instead of working with chromatic symmetric functions of graphs directly. We introduce a new combinatorial object: the correct sequences of unit interval orders, and, using these, we reprove monomial positivity of $G$-analogues of the power sum symmetric functions.