The Dehn–Sommerville relations and the Catalan matroid

Abstract

The f f -vector of a d d -dimensional polytope P P stores the number of faces of each dimension. When P P is a simplicial polytope the Dehn–Sommerville relations condense the f f -vector into the g g -vector, which has length ⌈ d + 1 2 ⌉ \lceil {\frac {d+1}{2}}\rceil . Thus, to determine the f f -vector of P P , we only need to know approximately half of its entries. This raises the question: Which ( ⌈ d + 1 2 ⌉ ) (\lceil {\frac {d+1}{2}}\rceil ) -subsets of the f f -vector of a general simplicial polytope are sufficient to determine the whole f f -vector? We prove that the answer is given by the bases of the Catalan matroid.