all of the same degree. We also note that in their study of pure $O$-sequences Boij, Migliore, Miró-Roig, the second author, and Zanello [@BMMNZ] have explored the relation between the weak Lefschetz property and pure $O$-sequences.

In this paper we extend the connection found by Chen, Guo, Jin, and Liu to a connection between artinian monomial almost complete intersections in three variables and lozenge tilings of more general regions that we call punctured hexagons. In Section \[sec:toolchain\] we gather a few useful tools for dealing with the weak Lefschetz property. In Section \[sec:aci\] we introduce the algebras we are interested in: artinian monomial almost complete intersections in three variables. If the syzygy bundle is not semistable, then the algebra has the weak Lefschetz property in characteristic zero ([@BK]). Thus we focus on the algebras that have semistable syzygy bundles, which we classify numerically (Proposition \[pro:semistable\]). Then we prove that such an algebra has the weak Lefschetz property if and only if a particular map between the peak homogeneous components of the algebra is a bijection (Corollary \[cor:one-map\]). Using this, we show that to each of the studied algebras $A$ we can associate a zero-one matrix $Z_A$ such that $A$ has 