On the shape of a pure O -sequence

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A monomial order ideal is a finite collection X of (monic) monomials such that, whenever M∈X and N divides M, then N∈X. Hence X is a poset, where the partial order is given by divisibility. If all, say tt, maximal monomials of X have the same degree, then X is pure (of type t).

A pure O-sequence is the vector, h=(h0=1,h1,...,he), counting the monomials of X in each degree. Equivalently, pure O-sequences can be characterized as the ff-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 ff-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:


A characterization of the first half of a pure O-sequence, which yields the exact converse to a g-theorem of Hausel;


A study of (the failing of) the unimodality property;


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;


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;


A pithy connection of the ICP with Stanley's conjecture on the h-vectors of matroid complexes;


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).


An analysis, from a commutative algebra viewpoint, of the extent to which the Weak and Strong Lefschetz Properties can fail for monomial algebras.


Some observations about pure f-vectors, an important special case of pure O-sequences.

Publisher's Statement

© Copyright 2012, American Mathematical Society. Publisher’s version of record: http://dx.doi.org/10.1090/S0065-9266-2011-00647-7