# Chromatic nonsymmetric polynomials of Dyck graphs are slide-positive

## TL;DR

This paper introduces a positive expansion for chromatic nonsymmetric polynomials of Dyck graphs in fundamental slide polynomials, connecting to Macdonald polynomials and extending known expansions of chromatic quasisymmetric functions.

## Contribution

It provides a new positive expansion in fundamental slide polynomials for chromatic nonsymmetric polynomials of Dyck graphs, utilizing flagged $(P,
ho)$-partitions and linking to existing expansions.

## Key findings

- Positive expansion in fundamental slide polynomials
- Connection to Macdonald polynomials
- Backstable limit recovers known expansions

## Abstract

Motivated by the study of Macdonald polynomials, J. Haglund and A. Wilson introduced a nonsymmetric polynomial analogue of the chromatic quasisymmetric function called the \emph{chromatic nonsymmetric polynomial} of a Dyck graph. We give a positive expansion for this polynomial in the basis of fundamental slide polynomials using recent work of Assaf-Bergeron on flagged $(P,\rho)$-partitions. We then derive the known expansion for the chromatic quasisymmetric function of Dyck graphs in terms of Gessel's fundamental basis by taking a backstable limit of our expansion.