# On the Shape of a Pure \(O\)-Sequence

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*Mats Boij; Juan C Migliore; Rosa M. Miró-Roig; Uwe Nagel; Fabrizio Zanello*

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.

#### Table of Contents

# Table of Contents

## On the Shape of a Pure $O$-Sequence

- Acknowledgments vii8 free
- Chapter 1. Introduction 110 free
- Chapter 2. Definitions and preliminary results 716 free
- Chapter 3. Differentiability and unimodality 1120
- Chapter 4. The Interval Conjecture for Pure O-sequences 1928
- Chapter 5. Enumerating pure O-sequences 2938
- Chapter 6. Monomial Artinian level algebras of type two in three variables 3948
- Chapter 7. Failure of the WLP and the SLP 5362
- Chapter 8. Remarks on pure f-vectors 6170
- Chapter 9. Some open or open-ended problems 6776
- Appendix A. Collection of definitions and notation 7180
- Bibliography 7584