  Let $\mathcal{A}$ be a Weyl arrangement. We introduce and study the notion of
$\mathcal{A}$-Eulerian polynomial producing an Eulerian-like polynomial for any
subarrangement of $\mathcal{A}$. This polynomial together with shift operator
describe how the characteristic quasi-polynomial of a new class of arrangements
containing ideal subarrangements of $\mathcal{A}$ can be expressed in terms of
the Ehrhart quasi-polynomial of the fundamental alcove. The method can also be
extended to define two types of deformed Weyl subarrangements containing the
families of the extended Shi, Catalan, Linial arrangements and to compute their
characteristic quasi-polynomials. We obtain several known results in the
literature as specializations, including the formula of the characteristic
polynomial of $\mathcal{A}$ via Ehrhart theory due to Athanasiadis (1996),
Blass-Sagan (1998), Suter (1998) and Kamiya-Takemura-Terao (2010); and the
formula relating the number of coweight lattice points in the fundamental
parallelepiped with the Lam-Postnikov Eulerian polynomial due to the third
author.
