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

We give a combinatorial formula for the Ehrhart $h^*$-vector of the hypersimplex. In particular, we show that $h^{*}_{d}(Δ_{k,n})$ 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.

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