2 DECOMPOSITIONS IN TOPOLOGICAL COMBINATORICS WITH APPLICATIONS TO MATROID COMPLEXES

This paper introduces two new decomposition techniques which are relat ed to the classical notion of shellability of simplicial complexes, an d uses the existence of these decompositions to deduce certain numeric al properties for an associated enumerative invariant. First, we intro duce the notion of M-shellability, which is a generalization to pure p osets of the property of shellability of simplicial complexes, and der ive inequalities that the rank-numbers of M-shellable posets must sati sfy. We also introduce a decomposition property for simplicial complex es called a convex ear-decomposition, and, using results of Kalai and Stanley on h-vectors of simplicial polytopes, we show that h-vectors o f pure rank-d simplicial complexes that have this property satisfy h(0 ) less than or equal to h(1) less than or equal to ... h([d/2]) and h( i) less than or equal to h(d-i) for 0 less than or equal to i less tha n or equal to (d/2). We then show that the abstract simplicial complex formed by the collection of independent sets of a matroid (or matroid complex) admits a special type of convex ear-decomposition called a P S ear-decomposition. This enables us to construct an associated M-shel lable poset, whose set of rank-numbers is the h-vector of the matroid complex. This results in a combinatorial proof of a conjecture of Hibi [17] that the h-vector of a matroid complex satisfies the above two s ets of inequalities.