Let $\mathcal A$ be a central hyperplane arrangement in $\mathbb R^d$ and let's assume that it is essential, meaning the hyperplanes in $\mathcal A$ intersect in the origin. The intersection lattice $L(\mathcal A)$ consists of those subspaces of $\mathbb R^d$ that are intersections of hyperplanes from $\mathcal A$ (including the empty intersection which is interpreted as $\mathbb R^d$) ordered by *reverse* inclusion. So notice that $\mathbb R^d$ is the bottom of $L(\mathcal A)$ and $\{0\}$ is the top. Let $\mu$ be the Mobius function of $L(\mathcal A)$.

Consider the quantity $$Q_{\mathcal A}=\sum_{W\in L(\mathcal A)}|\mu (W,\{0\})|.$$

Does $Q_{\mathcal A}$ have a natural interpretation in terms of the combinatorics of the arrangement? Note that Zaslavsky's theorem gives $$\sum_{W\in L(\mathcal A)}|\mu(\mathbb R^d,W)|$$ is the number of chambers of the arrangement, but this is, in some sense that is imprecise, a dual quantity.

I know that when $\mathcal A$ is the braid arrangement in $\mathbb R^n$, $Q_{\mathcal A}$ counts the number of necklaces of set partitions of $n$ (i.e., you look at ordered set partitions of $\{1,\ldots,n\}$ up to cyclic permutations).