$q$-Chromatic polynomials

We study a $q$-version of the chromatic polynomial of a given graph $G=(V,E)$, namely, $${\chi }_G^{\lambda }(q,n):=\sum _{\begin{array}{c}\text{proper colorings}\\ c:V\to [n]\end{array}}{q}^{\sum _{v\in V}{\lambda }_{v}c(v)},$$ where $\lambda \in {\mathbb{Z}}_{>0}^V$ is a fixed linear form. Via work of Chapoton (2016) on $q$-Ehrhart polynomials, ${\chi }_G^{\lambda }(q,n)$ turns out to be a polynomial in the $q$-integer $[n{]}_q$, with coefficients that are rational functions in $q$. Additionally, we prove structural results for ${\chi }_G^{\lambda }(q,n)$ and exhibit connections to neighboring concepts, e.g., chromatic symmetric functions and the arithmetic of order polytopes. We offer a strengthened version of Stanley’s conjecture that the chromatic symmetric function distinguishes trees, which leads to an analogue of $P$-partitions for graphs.