is additive with respect to disjoint union, we get the result.

From the half-open hypercube to the closed hypercube {#sec:inclexcl}
====================================================

The [*multi-hypersimplices*]{} are the polytopes defined by: $$\mathcal{B}^{(r)}_{n,k} :=  
     \begin{cases} 
 \big\{ v\in [0,r]^n \, : \, k-1 \leq \sum v_i < k       \big\}  & \text{ if } 1 \leq k < rn, \\[3mm]
 \big\{ v\in [0,r]^n \, : \, k-1 \leq \sum v_i \leq k    \big\}  & \text{ if } k=rn.
     \end{cases}$$

These polytopes form a particular class of the ones introduced by Lam and Postnikov [@lam] under the same name. The polytopes $\mathcal{B}^{(1)}_{n,k}$ are simply called the hypersimplices, and they can be described geometrically as truncated simplices (this fact is essentially due to Coxeter [@coxeter Section 8.7]).

Note that $$[0,r]^n = \biguplus_{ 1 \leq k \leq rn } \mathcal{B}^{(r)}_{n,k}.$$

\[relab\] Let $B^{(r)}_{n,k}(z) = E^*( \mathcal{B}^{(r)}_{n,k} , z ) $ and $A^{(r)}_{n,k}(z) = E^*( \mathcal{A}^{(r)}_{n,k} , z ) $, with the convention that $A_{n,0}^{(r)}(z) = B_{n,0}^{(r)}(z) = \delta_{n0}$ and $A_{n,k}^{(r)}(z) = B_{n,k}^{(r)}(z) = 0$ if $k<0$ or $k>rn$. Then: $$B^{(r)}_{n,k}(z) = \sum_{j=0}^n \binom nj  (1-z)^j A^{(r)}_{n-j,k-rj}(z).$$

For each $\Delta \subset \{1, \dots, n\}$, let $ \mathcal{H}_\Delta =  \{ v\in \mathcal{B}^{(r)}_{n,k} \, : \, v_i=r \text{ iff } i \in 