# Algebra and Geometry Seminar: Lattice Points in Slices of Prisms

**Author: Lona** | Image: Lona

**Author: Lona** | Image: Lona

**Abstract:** We study the Ehrhart polynomials of certain slices of rectangular prisms. 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 i.e. the coefficients of the Ehrhart polynomial are positive. Additionally, providing an extension of a result by Early and Kim, we give a combinatorial interpretation for all the coefficients of the

–polynomial, and hence we solve the problem of providing an interpretation for the numerator of the Hilbert series, also known as the h-vector of all algebras of Veronese type. As consequences of our results, we obtain an expression for the volumes of these slices of prisms as weighted combinations of Eulerian numbers; we use this to provide some generalizations of Laplace’s result on the combinatorial interpretation of the volume of the hypersimplex.