The full root polytope of type $A$ is the convex hull of all pairwise differences of the standard basis vectors which we represent by forward and backward arrows. We completely classify all flag triangulations of this polytope that are uniform in the sense that the edges may be described as a function of the relative order of the indices of the four basis vectors involved. These fifteen triangulations fall naturally into three classes: three in the lex class, three in the revlex class and nine in the Simion class. We also consider a refined face count where we distinguish between forward and backward arrows. We prove the refined face counts only depend on the class of the triangulations. The refined face generating functions are expressed in terms of the Catalan and Delannoy generating functions and the modified Bessel function of the first kind.

中文翻译：

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A型全根多胞体边界的统一旗三角剖分分类

$A$ 类型的完整根多胞形是我们用向前和向后箭头表示的标准基向量的所有成对差异的凸包。我们完全分类了这个多面体的所有标志三角剖分，这些三角剖分在边缘可以描述为所涉及的四个基向量的索引的相对顺序的函数的意义上是一致的。这十五个三角剖分自然地分为三类：三个在 lex 类中，三个在 revlex 类中，九个在 Simion 类中。我们还考虑了一个精细的面部计数，我们区分了向前和向后的箭头。我们证明了精致的人脸计数仅取决于三角剖分的类别。改进的人脸生成函数用 Catalan 和 Delannoy 生成函数以及第一类修正的 Bessel 函数表示。