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

Abstract

We give a combinatorial formula for the Ehrhart $h^{\ast }$-vector of the hypersimplex. In particular, we show that $h_d^{\ast }({\Delta }_{k,n})$ 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 $h^{\ast }$-vector of a generic cross-section of a hypercube.