SIGNABLE POSETS AND PARTITIONABLE SIMPLICIAL COMPLEXES

The notion of a partitionable simplicial complex is extended to that o f a signable partially ordered set. It is shown in a unified way that face lattices of shellable polytopal complexes, polyhedral cone fans, and oriented matroid polytopes, are all signable. Each of these classe s, which are believed to be mutually incomparable, strictly contains t he class of convex polytopes. A general sufficient condition, termed t otal signability, for a simplicial complex to satisfy McMullen's Upper Bound Theorem on the numbers of faces, is provided. The simplicial me mbers of each of the three classes above are concluded to be partition able and to satisfy the upper bound theorem. The computational complex ity of face enumeration and of deciding partitionability is discussed. It is shown that under a suitable presentation, the face numbers of a signable simplicial complex can be efficiently computed. In particula r, the face numbers of simplicial fans can be computed in polynomial t ime, extending the analogous statement for convex polytopes.