Abstract:
In a joint work with Migliore, Miró-Roig, Nagel and Zanello we have looked at the set of pure $O$-sequences. We can think of a pure $O$-sequence in several possible ways, for example as counting the monomials of various degrees of an order ideal of monomials where all the maximal monomials have the same degree or as the Hilbert function of a monomial Artinian level algebra. The concept was introduced by Stanley and Hibi gave a very general necessary condtion on such sequences. In our work, we propose that while the boundary of the set of pure $O$-sequences is known to be hard to characterize, we can still hope for a nice internal structure. We investigate how well-behaved pure $O$-sequences are with respect to unimodality. Moreover, we look at the weak Lefschetz property for monomial Artinian level algebras, since the presence of that provide further strong restrictions on the Hilbert function.