The Dehn-Sommerville Relations and the Catalan Matroid

The f-vector of a d-dimensional polytope P stores the number of faces of each dimension. When P is simplicial the Dehn--Sommerville relations condense the f-vector into the g-vector, which has length ⌈d+12⌉. Thus, to determine the f-vector of P, we only need to know approximately half of its entries. This raises the question: Which (⌈d+12⌉)-subsets of the f-vector of a general simplicial polytope are sufficient to determine the whole f-vector? We prove that the answer is given by the bases of the Catalan matroid.