#### Title

On the shape of a pure O -sequence

#### Document Type

Book

#### Publication Date

10-2011

#### Abstract

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=(h_{0}=1,h_{1},...,h_{e}), 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:

**(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.

#### ISBN

978-0-8218-9010-3

#### Recommended Citation

Boij, M.,
Migliore, J.,
Miro-Roig, R.,
Nagel, U.,
&
Zanello, F.
(2011).
On the shape of a pure O -sequence.
*
,
218*(1024)
http://dx.doi.org/10.1090/S0065-9266-2011-00647-7

Retrieved from: https://digitalcommons.mtu.edu/math-fp/37

## Publisher's Statement

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