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?