Lefschetz Properties and $h$-vectors of Graded Artinian Algebras

A standard graded algebra $A$ over a field is said to have the weak Lefschetz property (WLP) if it contains a degree one element $\ell$ such that multiplication by $\ell$ from one degree component of $A$ to the next always has maximal rank. In this case the Hilbert function or $h$-vector of $A$ is unimodal. This vector records the vector space dimensions of the graded components of $A$. Even in the case that the relations of $A$ are given by monomials such an algebra may fail the WLP or have a non-unimodal $h$-vector. Computer experiments suggests that algebras with these properties are rather rare. We discuss classes of randomly generated monomial algebras whose expected $h$-vectors are unimodal.