Some combinatorial properties of hexagonal lattices

In this paper, we consider the combinatorial properties of the hexagonal lattice. Let $e(n)$ be the number of $n$-element order ideals in a hexagonal lattice. We give the enumeration of $e(n)$ by showing a bijection between the order ideals and Schröder paths. Further, we get formulae for the flag $f$- and $h$-vectors of the hexagonal lattice

Finiteness theorems for matroid complexes with prescribed topology

It is known that there are finitely many simplicial complexes (up to isomorphism) with a given number of vertices. Translating to the language of $h$-vectors, there are finitely many simplicial complexes of bounded dimension with $h_1=k$ for any natural number $k$. In this paper we study the question at the other end of the $h$-vector: Are there only finitely many $(d-1)$-dimensional simplicial complexes with $h_d=k$ for any given $k$? The answer is no if we consider general complexes, but when focus on three cases coming from matroids: (i) independence complexes, (ii) broken circuit complexes, and (iii) order complexes of geometric lattices. We prove the answer is yes in cases (i) and (iii) and conjecture it is also true in case (ii).Comment: to appear in European Journal of Combinatoric