  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.
