论文标题
棱镜切片的格子点
Lattice points in slices of prisms
论文作者
论文摘要
我们对某些矩形棱镜的某些切片的Ehrhart理论进行了系统的研究。我们的多面是概括性化的概括，并包含在Lam和Postnikov引入的较大的多甲状腺功能上。此外，它们与满足强大交易所财产达到亲和力的多符合性的多符合性相吻合。我们为所有EHRHART系数提供了一个组合公式，就满足某些兼容性属性的加权排列数量而言。该结果证明了所有这些多面体都是Ehrhart阳性的。此外，通过早期和KIM的结果扩展，我们为$ H^*$ - 多项式的所有系数提供了组合解释。我们所有的结果都提供了对Hilbert功能的组合理解，以及Veronese类型的所有代数的$ H $ - 向量，到目前为止，这个问题仍然难以捉摸。讨论了各种应用，包括这些棱镜切片的表达式，作为欧拉数字的加权组合；拉普拉斯（Laplace）的一些扩展结果是对超刺体体积的组合解释的结果；标志Eulerian数字和改进的多元概括；以及所有均匀基质体的独立性多元化的Ehrhart阳性的简短证明。
We conduct a systematic study of the Ehrhart theory of certain slices of rectangular prisms. Our polytopes are generalizations of the hypersimplex and are contained in the larger class of polypositroids introduced by Lam and Postnikov; moreover, they coincide with polymatroids satisfying the strong exchange property up to an affinity. We give a combinatorial formula for all the Ehrhart coefficients in terms of the number of weighted permutations satisfying certain compatibility properties. This result proves that all these polytopes are Ehrhart positive. Additionally, via an extension of a result by Early and Kim, we give a combinatorial interpretation for all the coefficients of the $h^*$-polynomial. All of our results provide a combinatorial understanding of the Hilbert functions and the $h$-vectors of all algebras of Veronese type, a problem that had remained elusive up to this point. A variety of applications are discussed, including expressions for the volumes of these slices of prisms as weighted combinations of Eulerian numbers; some extensions of Laplace's result on the combinatorial interpretation of the volume of the hypersimplex; a multivariate generalization of the flag Eulerian numbers and refinements; and a short proof of the Ehrhart positivity of the independence polytope of all uniform matroids.
