their combinatorial meaning for special polytopes. Perhaps the most basic example is the unit hypercube $[0,1]^n$ which has the $n$th Eulerian polynomial as Ehrhart series, as will be detailed below. Although we do not use this language here, a general method to find the Ehrhart series of a polytope is to use unimodular shellable triangulations (as was done for example in [@li]) and it is essentially the idea behind what follows.

If $v=(v_1,\dots,v_n)\in\mathbb{R}^n$, let $$\operatorname{des}(v) = \#\{ i \, : 1\leq i \leq n-1, \; v_i > v_{i+1} \}.$$ We define the [*standardization*]{} $\operatorname{std}(v)$ of $v$ to be the unique permutation $\sigma\in\mathfrak{S}_n$ such that for all $i<j$ we have $v_i \leq v_j$ iff $\sigma_i<\sigma_j$. For each permutation $\sigma\in\mathfrak{S}_n$, let $$\mathcal{S}_{\sigma} = \{ v \in [0,1]^n \, : \, \operatorname{std}(v)=\sigma \}.$$

For example, one can check that if $x<y<z$, $\operatorname{std}(x,y,x,z,y,x,x)=1527634$. Note that the unit hypercube $[0,1]^n$ is the disjoint union of the subsets $\mathcal{S}_{\sigma}$ for $\sigma \in \mathfrak{S}_n $. Geometrically, each $\mathcal{S}_\sigma$ is a unit simplex where some facets are removed. So it is not a polytope in the usual sense; we should call it a “partially open” polytope. But note that Equations  and  make sense even when $\mathcal{X}$ is not 