Memoirs of the American Mathematical Society 2012; 78 pp; softcover Volume: 218 ISBN10: 0821869108 ISBN13: 9780821869109 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. 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 openended problems
 Appendix A. Collection of definitions and notation
 Bibliography
