A combinatorial formula for the Ehrhart $h^{*}$-vector of the hypersimplex

TL;DR

This paper provides a combinatorial formula for the Ehrhart $h^*$-vector of the hypersimplex, linking it to hypersimplicial decorated ordered set partitions and proving a conjecture by Nick Early.

Contribution

It introduces a new combinatorial formula for the Ehrhart $h^*$-vector of the hypersimplex, confirming a conjecture by Nick Early.

Findings

$h^{*}_{d}( riangle_{k,n})$ equals the number of hypersimplicial decorated ordered set partitions with winding number $d$.

Proves a conjecture relating Ehrhart $h^*$-vector of hypersimplices to combinatorial objects.

Extends the result to generic cross-sections of hypercubes.

Abstract

We give a combinatorial formula for the Ehrhart $h^*$-vector of the hypersimplex. In particular, we show that $h^{*}_{d}(\Delta_{k,n})$ is the number of hypersimplicial decorated ordered set partitions of type $(k,n)$ with winding number $d$, thereby proving a conjecture of Nick Early. We do this by proving a more general conjecture of Nick Early on the Ehrhart $h^*$-vector of a generic cross-section of a hypercube.