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A quasisymmetric function generalization of the chromatic symmetric function. (English) Zbl 1213.05133
Summary: 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.

##### MSC:
 05C31 Graph polynomials 05E05 Symmetric functions and generalizations
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