# LLT polynomials, chromatic quasisymmetric functions and graphs with cycles

## TL;DR

This paper explores the combinatorics of LLT polynomials and chromatic quasisymmetric functions using Dyck path models, extending to circular arc digraphs, and investigates their $e$-positivity properties and coefficients.

## Contribution

It introduces a Dyck path model for circular arc digraphs, extending the study of LLT polynomials and chromatic quasisymmetric functions to new graph families and explores their $e$-positivity conjectures.

## Key findings

- Parallel phenomena in $e$-positivity of LLT and chromatic quasisymmetric functions.
- Natural combinatorial interpretations for $e$-coefficients of line and cycle graphs.
- Extension of Dyck path models to circular arc digraphs.

## Abstract

We use a Dyck path model for unit-interval graphs to study the chromatic quasisymmetric functions introduced by Shareshian and Wachs, as well as vertical strip --- in particular, unicellular LLT polynomials. We show that there are parallel phenomena regarding $e$-positivity of these two families of polynomials. In particular, we give several examples where the LLT polynomials behave like a "mirror image" of the chromatic quasisymmetric counterpart. The Dyck path model is also extended to circular arc digraphs to obtain larger families of polynomials. This circular extensions of LLT polynomials has not been studied before. A lot of the combinatorics regarding unit interval graphs carries over to this more general setting, and we prove several statements regarding the $e$-coefficients of chromatic quasisymmetric functions and LLT polynomials. In particular, we believe that certain $e$-positivity conjectures hold in all these families above. Furthermore, we study vertical-strip LLT polynomials, for which there is no natural chromatic quasisymmetric counterpart. These polynomials are essentially modified Hall--Littlewood polynomials, and are therefore of special interest. In this more general framework, we are able to give a natural combinatorial interpretation for the $e$-coefficients for the line graph and the cycle graph, in both the chromatic and the LLT setting.