| eBook ISBN: | 978-0-8218-9010-3 |
| Product Code: | MEMO/218/1024.E |
| List Price: | $60.00 |
| MAA Member Price: | $54.00 |
| AMS Member Price: | $36.00 |
| eBook ISBN: | 978-0-8218-9010-3 |
| Product Code: | MEMO/218/1024.E |
| List Price: | $60.00 |
| MAA Member Price: | $54.00 |
| AMS Member Price: | $36.00 |
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 218; 2012; 78 ppMSC: Primary 13; 05; 06; Secondary 14
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.
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Table of Contents
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Chapters
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1. Introduction
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2. Definitions and preliminary results
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3. Differentiability and unimodality
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4. The Interval Conjecture for Pure $O$-sequences
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5. Enumerating pure $O$-sequences
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6. Monomial Artinian level algebras of type two in three variables
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7. Failure of the WLP and the SLP
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8. Remarks on pure $f$-vectors
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9. Some open or open-ended problems
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A. Collection of definitions and notation
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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.
-
Chapters
-
1. Introduction
-
2. Definitions and preliminary results
-
3. Differentiability and unimodality
-
4. The Interval Conjecture for Pure $O$-sequences
-
5. Enumerating pure $O$-sequences
-
6. Monomial Artinian level algebras of type two in three variables
-
7. Failure of the WLP and the SLP
-
8. Remarks on pure $f$-vectors
-
9. Some open or open-ended problems
-
A. Collection of definitions and notation
