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On the Shape of a Pure O-sequence

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, p...