Eulerian numbers $A_{n,k}$ satisfy $$\frac{A_{n,k}}{n!} = {\rm Vol}(  \big\{ v\in [0,1]^n \, : \,  k-1 \leq \sum v_i \leq k  \big\} ),$$ this is essentially a calculation due to Laplace (see [@foata] for details). But the combinatorial definition can be easily translated in the following way: $$\frac{A_{n,k}}{n!} = {\rm Vol}(   \{ v\in [0,1]^n \, : \,  {\rm des}(v) = k-1  \} ).$$ The problem is to find a measure-preserving bijection between the two sets, to explain why they have the same volume. A simple solution was given by Stanley [@stanley].

The set $\{ v \in [0,1]^n \, : \, k\leq \sum v_i \leq k+1 \}$ is in fact a convex integral polytope known as the hypersimplex, and in this context we can consider the Ehrhart series, which is a generalization of the volume. This led to a recent conjecture by Stanley about the Ehrhart series of the hypersimplex (more precisely, a partially open version of the hypersimplex), which was proved by Nan Li [@li] in two different ways. It is remarkable that two Eulerian statistics are needed to state the conjecture, which says that the Ehrhart series of a hypersimplex is the descent generating function for permutations with a given 