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Combinatorial reciprocity for the chromatic polynomial and the chromatic symmetric function

By Olivier Bernardi and Philippe Nadeau

Abstract

Let G be a graph, and let χG be its chromatic polynomial. For any non-negative integers i, j, we give an interpretation for the evaluation χ (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

Topics: [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO]
Publisher: HAL CCSD
Year: 2019
OAI identifier: oai:HAL:hal-02086961v1

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