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

TL;DR

This paper proposes conjectures providing combinatorial interpretations of Ehrhart $h^*$-vectors for hypersimplices, dilated simplices, and cube cross-sections using decorated ordered set partitions, supported by computational evidence.

Contribution

It introduces new conjectures linking Ehrhart $h^*$-vectors to decorated ordered set partitions for various polytopes, expanding combinatorial understanding.

Findings

Conjectures formulated for hypersimplices, dilated simplices, and cube cross-sections.

Computational checks support the validity of the conjectures.

Provides a new combinatorial framework for Ehrhart $h^*$-vectors.

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.