Two unfortunate properties of pure $f$-vectors

Abstract

The set of f-vectors of pure simplicial complexes is an important but little understood object in combinatorics and combinatorial commutative algebra. Unfortunately, its explicit characterization appears to be a virtually intractable problem, and its structure very irregular and complicated. The purpose of this note, where we combine a few different algebraic and combinatorial techniques, is to lend some further evidence to this fact. We first show that pure (in fact, Cohen-Macaulay) f-vectors can be nonunimodal with arbitrarily many peaks, thus improving the corresponding results known for level Hilbert functions and pure O-sequences. We provide both an algebraic and a combinatorial argument for this result. Then, answering negatively a question of the second author and collaborators posed in the recent AMS Memoir on pure O-sequences, we show that the Interval Property fails for the set of pure f-vectors, even in dimension 2.