Seeking combinatorial interpretation of a quantity that comes up from central hyperplane arrangements

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).

• You mean "central," not "essential." May 14 '21 at 20:57
• @RichardStanley I meant both but only so I could call the intersection of the hyperplanes 0. Maybe my terminology is slightly off May 14 '21 at 21:16

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.
• See the above reference. For general simplicial complexes, the Betti numbers are sums of dimensions of homology groups of restrictions $\Delta_W$ of $\Delta$ to some subset $W$ of the vertices (Theorem 4). For matroid complexes, the homology of each $\Delta_W$ vanishes except in the top dimension (this property is in fact equivalent to being a matroid complex), so the top dimensional homology is just the reduced Euler characteristic of $\Delta_W$, up to sign. May 14 '21 at 23:27