The Baldoni–Vergne volume and Ehrhart polynomial formulas for flow polytopes are significant in at least two ways. On one hand, these formulas are in terms of Kostant partition functions, connecting flow polytopes to this classical vector partition function, fundamental in representation theory. On the other hand, the Ehrhart polynomials can be read off from the volume functions of flow polytopes. The latter is remarkable since the leading term of the Ehrhart polynomial of an integer polytope is its volume! Baldoni and Vergne proved these formulas via residues. To reveal the geometry of these formulas, the second author and Morales gave a fully geometric proof for the volume formula and a partial generating function proof for the Ehrhart polynomial formula. The goal of the present paper is to provide a fully geometric proof for the Ehrhart polynomial formula for flow polytopes.

流动多表位的Baldoni–Vergne体积和Ehrhart多项式公式至少在两种方面很重要。一方面，这些公式是用Kostant分区函数来表示的，将流多面体连接到这种经典的矢量分区函数，这是表示论的基础。另一方面，可以从流多面体的体积函数中读取Ehrhart多项式。后者之所以引人注目，是因为整数多面体的Ehrhart多项式的首项就是它的体积！Baldoni和Vergne通过残基证明了这些分子式。为了揭示这些公式的几何形状，第二作者和Morales为体积公式提供了完整的几何证明，并为Ehrhart多项式公式提供了部分生成函数的证明。