A Noncommutative Chromatic Symmetric Function

TL;DR

This paper introduces a noncommutative symmetric function that generalizes Stanley's chromatic symmetric function, satisfying a deletion-contraction law, enabling new proofs and progress on the Stanley-Stembridge conjecture.

Contribution

It defines a noncommutative symmetric function generalizing X_G, facilitating uniform proofs and advancing the (3+1)-free Conjecture.

Findings

Established a noncommutative symmetric function with deletion-contraction law

Generalized Stanley's theorems using the new function

Made progress on the Stanley-Stembridge (3+1)-free Conjecture

Abstract

Stanley associated with a graph G a symmetric function X_G which reduces to G's chromatic polynomial under a certain specialization of variables. He then proved various theorems generalizing results about the chromatic polynomial, as well as new ones that cannot be interpreted at that level. Unfortunately, X_G does not satisfy a Deletion-Contraction Law which makes it difficult to apply induction. We introduce a symmetric function in noncommuting variables which does have such a law and specializes to X_G when the variables are allowed to commute. This permits us to further generalize some of Stanley's theorems and prove them in a uniform and straightforward manner. Furthermore, we make some progress on the (3+1)-free Conjecture of Stanley and Stembridge.