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On the Shape of a Pure $$O$$-Sequence
Mats Boij, Royal Institute of Technology, Stockholm, Sweden, Juan C. Migliore, University of Notre Dame, IN, Rosa M. Miró-Roig, University of Barcelona, Spain, Uwe Nagel, University of Kentucky, Lexington, KY, and Fabrizio Zanello, Michigan Technological University, Houghton, MI
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Memoirs of the American Mathematical Society
2012; 78 pp; softcover
Volume: 218
ISBN-10: 0-8218-6910-8
ISBN-13: 978-0-8218-6910-9
List Price: US$60 Individual Members: US$36
Institutional Members: US\$48
Order Code: MEMO/218/1024

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.

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