eBook ISBN:  9780821890103 
Product Code:  MEMO/218/1024.E 
List Price:  $60.00 
MAA Member Price:  $54.00 
AMS Member Price:  $36.00 
eBook ISBN:  9780821890103 
Product Code:  MEMO/218/1024.E 
List Price:  $60.00 
MAA Member Price:  $54.00 
AMS Member Price:  $36.00 

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.

Table of Contents

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 openended problems

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 openended problems

A. Collection of definitions and notation