# Combinatorial reciprocity for the chromatic polynomial and the chromatic symmetric function

## TL;DR

This paper provides combinatorial interpretations for evaluations of the chromatic polynomial and symmetric function of a graph, extending classical results and employing heap theory for proofs.

## Contribution

It introduces new combinatorial interpretations for chromatic polynomial evaluations and extends these to symmetric functions, using heap theory.

## Key findings

- Interpretations for $inom{ ext{chromatic polynomial}}{i,j}$ evaluations
- Symmetric function refinements of these interpretations
- Extensions of classical combinatorial results

## Abstract

Let G be a graph, and let $\chi$G be its chromatic polynomial. For any non-negative integers i, j, we give an interpretation for the evaluation $\chi$ (i) G (--j) in terms of acyclic orientations. This recovers the classical interpretations due to Stanley and to Green and Zaslavsky respectively in the cases i = 0 and j = 0. We also give symmetric function refinements of our interpretations, and some extensions. The proofs use heap theory in the spirit of a 1999 paper of Gessel.