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.

中文翻译：

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外尔排列子排列的欧拉多项式

令 $\mathcal{A}$ 是 Weyl 排列。我们介绍并研究了 $\mathcal{A}$-欧拉多项式的概念，为 $\mathcal{A}$ 的任何子排列产生类似欧拉的多项式。该多项式与移位运算符一起描述了包含 $\mathcal{A}$ 的理想子排列的一类新排列的特征拟多项式如何可以用基本壁龛的 Ehrhart 拟多项式表示。该方法还可以扩展到定义包含扩展 Shi、Catalan、Linial 排列族的两种变形 Weyl 子排列，并计算它们的特征拟多项式。我们在文献中获得了几个已知的结果作为专业化，包括 Athanasiadis (1996) 通过 Ehrhart 理论得到的 $\mathcal{A}$ 特征多项式的公式，Blass-Sagan (1998)、Suter (1998) 和 Kamiya-Takemura-Terao (2010)；以及第三作者提供的与基本平行六面体中的配重格点数与Lam-Postnikov欧拉多项式相关的公式。