  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.
