The face vector of a half-open hypersimplex

The half-open hypersimplex ${\mathrm{\Delta }}_{n,k}^{\mathrm{\prime }}$ consists of those $x = (x_{1}, \ldots, x_{n}) \in[0,1]^n$with$k-1<x_1+\cdots+x_n\le k$,where$0 < k \leq n$\mathrm{.}The$f$-vector of a half-open hypersimplex and related generating functions are explicitly studied.