A symmetric function of increasing forests

TL;DR

This paper introduces a symmetric function related to increasing forests in indifference graphs, establishing linear relations shared with known functions and providing a combinatorial interpretation of LLT polynomial coefficients.

Contribution

It defines a new symmetric function for increasing forests and connects it to existing functions, offering new combinatorial insights into LLT polynomials.

Findings

The symmetric function satisfies specific linear relations.

Provides a combinatorial interpretation of LLT polynomial coefficients.

Strengthens previous descriptions of LLT polynomials.

Abstract

For an indifference graph $G$ we define a symmetric function of increasing spanning forests of $G$. We prove that this symmetric function satisfies certain linear relations, which are also satisfied by the chromatic quasisymmetric function and unicellular LLT polynomials. As a consequence we give a combinatorial interpretation of the coefficients of the LLT polynomial in the elementary basis (up to a factor of a power of $(q-1)$), strengthening the description given by Alexandersson and Sulzgruber.