A quasisymmetric function generalization of the chromatic symmetric function

Brandon Humpert

Published 2010-04-15, updated 2011-01-03Version 2

The chromatic symmetric function $X_G$ of a graph $G$ was introduced by Stanley. In this paper we introduce a quasisymmetric generalization $X^k_G$ called the $k$-chromatic quasisymmetric function of $G$ and show that it is positive in the fundamental basis for the quasisymmetric functions. Following the specialization of $X_G$ to $\chi_G(\lambda)$, the chromatic polynomial, we also define a generalization $\chi^k_G(\lambda)$ and show that evaluations of this polynomial for negative values generalize a theorem of Stanley relating acyclic orientations to the chromatic polynomial.