Let $n$ be a positive natural number. For all $\emptyset \subset S \subseteq \{1, \ldots, n\}$ and $k \in \mathbb{Z}$, define the hyperplane $H(S,k)$ in $\mathbb{R}^n$ given by the equations $$H(S,k):= \left\{ \sum_{i\in S} x_i = k \right\} $$
(The hyperplane arrangement obtained by taking the collection of $H(S,0)$ in $\mathbb{R}^n$ is called the resonance hyperplane arrangement. Counting its number of polytopes in $\mathbb{R}^n$ is the subject of this MO question by Paul Johnson: A natural refinement of the $A_n$ arrangement is to consider all $2^n-1$ hyperplanes given by the sums of the coordinate functions. Have you seen this arrangement? Is it completely intractable?)
Question: How many polytopes are there in the complement of the collection of all hyperplanes $H(S,k)$ in the hypercube $[0,1]^n$?
The only relevant hyperplanes are those with $0<k<|S|$, as the others do not intersect the interior of the hypercube. For example, the cases $|S|=1$ with $k=0,1$ define the faces of the hypercube.
When $n=2$ this is simply the unit square $\{ 0 \leq x_1, x_2 \leq 1\}$ divided in two polytopes by the equation $x_1 + x_2 =1$.
When $n=3$ the relevant hyperplanes are $x_1+x_2+x_3=1,2$ and $x_i+x_j=1$ for all $0 < i < j \leq 3$. The first two equations split the cube in $3$ regions, two simplices and a central polytope. The remaining three equations split the central polytope in $2^3=8$ polytopes, so the answer to my question is $10$.
Does anybody know a reference for this problem for general $n$, or can advise me on how to proceed? Thanks in advance!
(I have asked the same on Math Stackexchange: https://math.stackexchange.com/questions/2268404/hypercube-subdivision).
