Chromatic symmetric functions from the modular law

TL;DR

This paper introduces an algorithm for computing chromatic quasisymmetric functions of indifference graphs using the modular law, revealing positivity and unimodality properties and exploring coefficient concavity.

Contribution

It provides a novel algorithm based on the modular law applicable to a broad class of functions, including unicellular LLT polynomials, and characterizes coefficient properties in special graph cases.

Findings

Coefficients are positive unimodal polynomials in certain cases.

The algorithm applies to functions satisfying the modular law.

Logarithmic concavity of coefficients is discussed.

Abstract

In this article we show how to compute the chromatic quasisymmetric function of indifference graphs from the modular law introduced by Guay-Paquet. We provide an algorithm which works for any function that satisfies this law, such as unicellular LLT polynomials. When the indifference graph has bipartite complement it reduces to a planar network, in this case, we prove that the coefficients of the chromatic quasisymmetric function in the elementary basis are positive unimodal polynomials and characterize them as certain $q$-hit numbers (up to a factor). Finally, we discuss the logarithmic concavity of the coefficients of the chromatic quasisymmetric function.