Conjectures for Ehrhart $h^*$-vectors of Hypersimplices and Dilated Simplices

Abstract

We formulate conjectures giving combinatorial interpretations of the Ehrhart $h^*$-vector, for hypersimplices, for dilated simplices and for generic cross-sections of cubes, in terms of certain decorated ordered set partitions. All were formulated and checked computationally during our graduate study at Penn State.