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).



Seeking combinatorial interpretation of a quantity that comes up from central hyperplane arrangements
Let $\Delta$ be a matroid complex, i.e., an abstract simplicial complex whose faces are the independent sets of a matroid $M$. Let $K[\Delta]$ denote the face ring (aka "Stanley-Reisner ring") of $\Delta$ over a field $K$. Let $\beta\_i(K[\Delta])$ denote the Betti numbers of a minimal free resolution of $K[\Delta]$, regarded as a module (in fact, quotient ring) over $K[x\_1,\dots,x\_n]$, where $x\_1,\dots,x\_n$ are the vertices of $\Delta$. Then
$$ \beta\_i(K[\Delta]) = \sum |\mu(W,\{0\})|, $$
where $W$ ranges over all flats of $M$ of corank $i$. In particular, the sum of all the Betti numbers is the sum requested. See Theorem 9 of [http://math.mit.edu/~rstan/pubs/pubfiles/34.pdf](http://math.mit.edu/%7Erstan/pubs/pubfiles/34.pdf).


