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On the shape of a pure $O$-sequence

About this Title

Mats Boij, Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden, Juan C. Migliore, Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, Rosa M. Miró-Roig, Facultat de Matemàtiques, Department d’Àlgebra i Geometria, Gran Via des les Corts Catalanes 585, 08007 Barcelona, Spain, Uwe Nagel, Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lexington, Kentucky 40506-0027 and Fabrizio Zanello, Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931

Publication: Memoirs of the American Mathematical Society
Publication Year: 2012; Volume 218, Number 1024
ISBNs: 978-0-8218-6910-9 (print); 978-0-8218-9010-3 (online)
Published electronically: October 3, 2011
Keywords:Pure $O$-sequence, Artinian algebra, monomial algebra, unimodality, differentiable $O$-sequence, level algebra, Gorenstein algebra, enumeration, interval conjecture, $g$-element, weak Lefschetz property, strong Lefschetz property, matroid simplicial complex, Macaulay’s inverse system.
MSC: Primary 13D40, 05E40, 06A07, 13E10, 13H10; Secondary 05A16, 05B35, 14M05, 13F20

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Table of Contents


  • Chapter 1. Introduction
  • Chapter 2. Definitions and preliminary results
  • Chapter 3. Differentiability and unimodality
  • Chapter 4. The Interval Conjecture for Pure $O$-sequences
  • Chapter 5. Enumerating pure $O$-sequences
  • Chapter 6. Monomial Artinian level algebras of type two in three variables
  • Chapter 7. Failure of the WLP and the SLP
  • Chapter 8. Remarks on pure $f$-vectors
  • Chapter 9. Some open or open-ended problems
  • Appendix A. Collection of definitions and notation


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:

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.

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