  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.
