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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?

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  • $\begingroup$ This is a trivial observation, but if you use the finite field method, you get the characteristic polynomial is $\chi(q) = q^n - |\{(a_1,\ldots,a_n) \in \mathbb{F}_q^n\colon \textrm{some nonempty subset of the $a_i$ sums to $0 \mod q$}\}|$. That looks like a terribly difficult counting problem, of course. $\endgroup$ – Sam Hopkins Jan 15 '13 at 17:06
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This is not known as far as I know, and seems to be a hard problem, see the references and comments in this MO question.

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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.

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