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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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