On the shape of a pure O-sequence

Abstract

A monomial order ideal is a finite collection $ X$ of (monic) monomials such that, whenever $ M\in X$ and $ N$ divides $ M$, then $ N\in X$. Hence $ X$ is a poset, where the partial order is given by divisibility. If all, say $ t$, maximal monomials of $ X$ have the same degree, then $ X$ is pure (of type $ t$). A pure $ O$-sequence is the vector, $ \underline {h}=(h_0=1,h_1,...,h_e)$, counting the monomials of $ X$ in each degree. Equivalently, pure $ O$-sequences can be characterized as the $ f$-vectors of pure multicomplexes, or, in the language of commutative algebra, as the $ h$-vectors of monomial Artinian level algebras. Pure $ O$-sequences had their origin in one of the early works of Stanley's in this area, and have since played a significant role in at least three different disciplines: the study of simplicial complexes and their $ f$-vectors, the theory of level algebras, and the theory of matroids. This monograph is intended to be the first systematic study of the theory of pure $ O$-sequences. Our work, which makes an extensive use of both algebraic and combinatorial techniques, in particular includes: (i) A characterization of the first half of a pure $ O$-sequence, which yields the exact converse to a $ g$-theorem of Hausel; (ii) A study of (the failing of) the unimodality property; (iii) The problem of enumerating pure $ O$-sequences, including a proof that almost all $ O$-sequences are pure, a natural bijection between integer partitions and type 1 pure $ O$-sequences, and the asymptotic enumeration of socle degree 3 pure $ O$-sequences of type $ t$; (iv) A study of the Interval Conjecture for Pure $ O$-sequences (ICP), which represents perhaps the strongest possible structural result short of an (impossible?) full characterization; (v) A pithy connection of the ICP with Stanley's conjecture on the $ h$-vectors of matroid complexes; (vi) A more specific study of pure $ O$-sequences of type 2, including a proof of the Weak Lefschetz Property in codimension 3 over a field of characteristic zero. As an immediate corollary, pure $ O$-sequences of codimension 3 and type 2 are unimodal (over an arbitrary field). (vii) An analysis, from a commutative algebra viewpoint, of the extent to which the Weak and Strong Lefschetz Properties can fail for monomial algebras. (viii) Some observations about pure $ f$-vectors, an important special case of pure $ O$-sequences.