  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.
