An equivalent formulation of chromatic quasi-polynomials

TL;DR

This paper proves that two different types of quasi-polynomials related to arrangements and colorings are actually equivalent, unifying their interpretations and applications in combinatorics.

Contribution

It establishes the equivalence between Chen-Wang's quasi-polynomial and Brändén-Moci's chromatic quasi-polynomial, showing they count the same combinatorial objects.

Findings

The two quasi-polynomials are mathematically equivalent.

They enumerate the same set sizes in different contexts.

This unification simplifies the understanding of related combinatorial invariants.

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.