TITLE: connected components of a real hyperplane arrangement
QUESTION [5 upvotes]: Let us consider the following configuration of hyperplanes in the real
vector space V with coordinates $z_1,\ldots,z_n$: the hyperplanes are
numbered by all the nonempty subsets $J\subset I=\{1,\ldots,n\}$, and the
hyperplane $H_J$ is given by $\sum_{i\in J}z_i=0$.
Question: How many connected components does the complement
$V\setminus\cup_{J}H_J$ have? That is, what are they naturally numbered by?

REPLY [3 votes]: There is a formula for the number of components which unfortunately is pretty useless,
namely, 
  $$ \sum_{k,r} \frac{(-1)^{n+k+r}}{k!}f(k,n,r), $$
where $f(k,n,r)$ is the number of real $k\times n$ $(0,1)$-matrices of rank $r$ with no zero row and no two rows equal.