# Number of regions of a hyperplane arrangement avoiding a generic hyperplane

Let $\mathcal{A}$ be an essential arrangement of hyperplanes in $\mathbb{R}^n$. Zaslavsky's theorem says that the number of regions of $\mathcal{A}$ is given by $r(\mathcal{A})=(-1)^n\chi_{\mathcal{A}}(-1)$ and the number of bounded regions is given by $b(\mathcal{A})=(-1)^n\chi_{\mathcal{A}}(1)$. (See Theorem 2.5 of Stanley's notes on hyperplane arrangements: http://math.mit.edu/~rstan/arrangements/arr.html).

I am interested in the following result which counts something "between" the number of regions and number of bounded regions of $\mathcal{A}$. Namely, let us say that the hyperplane $H$ is generic with respect to $\mathcal{A}$ if for any $H_1,\ldots,H_m \in \mathcal{A}$, we have that $H$ has nonempty intersection with $H_1 \cap \ldots \cap H_m$ if and only if $\mathrm{dim}(H_1 \cap \ldots \cap H_m) \geq 1$. Suppose $H$ is generic with respect to $\mathcal{A}$. Then the number of regions $R$ of $\mathcal{A}$ with $R \cap H = \emptyset$ is given by $(-1)^n\chi_{\mathcal{A}}(0)$.

It is not hard to deduce this fact from Zaslavsky's theorem. But it seems basic enough that I suspect it must've been observed before: so I am asking for a reference to anywhere this result has been stated.

## 1 Answer

Sorry to answer my own question, but I found a reference. See Theorem 3.1 of Greene and Zaslavsky's "On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions, and orientations of graphs", Trans. AMS 1983, available for free at http://www.ams.org/journals/tran/1983-280-01/S0002-9947-1983-0712251-1/.