Since the 1970's, great interest has been taken in the study of pure $O$-sequences, which, due to Macaulay's theory of inverse systems, have a bijective correspondence to the Hilbert functions of Artinian level monomial algebras. Much progress has been made in classifying these according to their shape. Macaulay's theorem immediately gives us that all Artinian algebras in two variables have unimodal Hilbert functions. Furthermore, it has been shown that all Artinian level monomial algebras of type two in three variables have unimodal Hilbert functions. This paper will classify all Artinian level monomial algebras of type two in four variables into two classes of ideals, prove that they are strictly unimodal and show that one of the classes is licci.
"The unimodality of pure $O$-sequences of type two in four variables." Rocky Mountain J. Math. 45 (6) 1781 - 1799, 2015. https://doi.org/10.1216/RMJ-2015-45-6-1781