A combinatorial perspective on the non-Radon partitions

Let E be a finite set of points in $\text{R}$^d. Then {A, E ? A} is a non-Radon partition of E iff there is a hyperplane H separating A strictly from E?A. Or equivalently iff ${}_{\mathrm{<mtext>A</mtext>}}\text{O}$ is an acyclic reorientation of (M_Aff(E), O), the oriented matroid canonically determined by E. If (M(E), O) is an oriented matroid without loops then the set $\text{NR(E, O) = {(A, E ? A):}{}_{\mathrm{<mtext>A</mtext>}}\text{O}\text{is acyclic}\text{}}$ determines (M(E), O). In particular the matroidal properties of a finite set of points in $\text{R}$^d are precisely the properties which can be formulated in non-Radon partitions terms. The Möbius function of the poset $\text{A = {A: A ? E,}{}_{\mathrm{<mtext>A</mtext>}}\text{O}\text{is acyclic}\text{}}$ and in a special case its homotopy type are computed. This paper generalizes recent results of P. Edelman (A partial order on the regions of $\text{R}$ⁿ dissected by hyperplanes