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We conduct a systematic study of the Ehrhart theory of certain slices ofrectangular prisms. Our polytopes are generalizations of the hypersimplex andare contained in the larger class of polypositroids introduced by Lam andPostnikov; moreover, they coincide with polymatroids satisfying the strongexchange property up to an affinity. We give a combinatorial formula for allthe Ehrhart coefficients in terms of the number of weighted permutationssatisfying certain compatibility properties. This result proves that all thesepolytopes are Ehrhart positive. Additionally, via an extension of a result byEarly and Kim, we give a combinatorial interpretation for all the coefficientsof the $h^*$-polynomial. All of our results provide a combinatorialunderstanding of the Hilbert functions and the $h$-vectors of all algebras ofVeronese type, a problem that had remained elusive up to this point. A varietyof applications are discussed, including expressions for the volumes of theseslices of prisms as weighted combinations of Eulerian numbers; some extensionsof Laplace's result on the combinatorial interpretation of the volume of thehypersimplex; a multivariate generalization of the flag Eulerian numbers andrefinements; and a short proof of the Ehrhart positivity of the independencepolytope of all uniform matroids.

Lattice points in slices of prisms