algebras which [*always*]{} have the weak Lefschetz property, regardless of the field characteristic (Corollary \[cor:C-zero-unique\]); we also describe a different subfamily of algebras which always have the weak Lefschetz property (Proposition \[pro:gamma-zero-unique\]). Moreover, we describe the unique algebras which retain certain properties yet have minimal multiplicity (Example \[exa:minimal-multiplicity\]).

In Section \[sec:splitting-type\] we explicitly determine (in one case, depending on the presence of the weak Lefschetz property) the splitting type of all artinian monomial almost complete intersections. In particular, we consider separately the cases when the syzygy bundle is non-semistable (Proposition \[pro:st-nss\]) and semistable (Propositions \[pro:st-nmod3\] and \[pro:st-mod3\]). Moreover, in the case of ideals associated to punctured hexagons, we relate the weak Lefschetz property to a number of other problems in algebra, combinatorics, and algebraic geometry (Theorem \[thm:equiv\]).

Finally, in Appendix \[sec:hyper-calculus\] we provide a technique, a “picture-calculus”, for working with hyperfactorials, a basic unit for the aforementioned closed formulae. We demonstrate that several nice polynomials can be written as ratios of products of hyperfactorials (Proposition \[pro:hyper-f\] and Corollary \[cor:hyper-eo-f\]). Further, this shows that MacMahon’s formula for the number of lozenge tilings of a (non-punctured) hexagon is a polynomial in one of the side-lengths when the other two are fixed (Corollary 