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arXiv · 1803.08649

An equivalent formulation of chromatic quasi-polynomials

Tan Nhat Tran
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Abstract

Given a central integral arrangement, the reduction of the arrangement modulo positive integers $q$ gives rise to a subgroup arrangement in $(\mathbb{Z}/q\mathbb{Z})^\ell$. Kamiya-Takemura-Terao (2008) introduced the notion of characteristic quasi-polynomials, which uses to evaluate the cardinality of the complement of the subgroup arrangement. Chen-Wang (2012) found a similar but more general setting that replacing the integral arrangement by its restriction to a subspace of $\mathbb{R}^\ell$, and evaluating the cardinality of the $q$-reduction complement will also lead to a quasi-polynomial in $q$. On an independent study, Br\"and\'en-Moci (2014) defined the so-called chromatic quasi-polynomial, and initiated the study of $q$-colorings on a finite list of elements in a finitely generated abelian group. The main purpose of this paper is to verify that the Chen-Wang's quasi-polynomial and the Br\"and\'en-Moci's chromatic quasi-polynomial are equivalent in the sense that the quasi-polynomials enumerate the cardinalities of isomorphic sets.

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Tan Nhat Tran. 2018-03-23. An equivalent formulation of chromatic quasi-polynomials. https://doi.org/10.1016/j.disc.2020.112012

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