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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
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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

  • 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
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