Acyclic reorientations of weakly oriented matroids

Theorem A extends to oriented matroids a theorem for graphs due to Stanley [ 131. It contains Zaslavsky’s [ 141 result, published independently the same year, on the number of regions determined by hyperplanes in Rd. Generalizations of Theorem A can be found in [6, 7, 10-121. The number a(M) = t(M; 2,0) is an important invariant of an oriented matroid M. By Theorem A, a(M) counts the number of acyclic reorientations of M. As follows easily from the oriented matroid generalization of Farkas’ lemma [2, Thm. 2.21, there is a l-l correspondence between acyclic reorientations and maximal covectors of a loopless oriented matroid M (a maximal covector of M is an inclusion-maximal signed span of M* in the terminology of [2, Sect. 51). Hence a(M) counts the number of maximal covectors of M. The construction in the proof of the Folkman-Lawrence [4] representation theorem establishes a l-l correspondence between maximal covectors of M and regions of its topological representation. Hence a(M) counts the number of regions of the Folkman-Lawrence topological representation.