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 $\lceil \frac{d+1}{2}\rceil$. Thus, to determine the $f$-vector of $P$, we only need to know approximately half of its entries. This raises the question: Which $(\lceil \frac{d+1}{2}\rceil )$-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.