On the shape of a pure $O$-sequence

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.