The Dehn-Sommerville Relations and the Catalan Matroid

TL;DR

This paper explores how the $f$-vector of a simplicial polytope can be reconstructed from a minimal subset of entries, revealing that the bases of the Catalan matroid are precisely what is needed.

Contribution

It establishes a novel connection between the $f$-vector reconstruction problem and the bases of the Catalan matroid, providing a combinatorial characterization.

Findings

The $f$-vector of a simplicial polytope can be determined from specific subsets.

Catalan matroid bases characterize the minimal subsets needed.

The result links polytope face enumeration with matroid theory.

Abstract

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.