notation, one immediate consequence ([@HMNW Remark 3.3]) of $A$ having the weak Lefschetz property is that the $h$-vector of $A$ is [*strictly unimodal*]{}. Further, the positive part of the first difference of $h(A)$ is $h(A/\ell A)$, where $\ell$ is any Lefschetz element of $A$.

The weak Lefschetz property is known to be subtle to both deformations (see, e.g., [@CN-2010], [@Mi], and [@MMN]) but also to field characteristic. The latter, considering the weak Lefschetz property in positive characteristic, is an exciting and active direction of research. Migliore, Miró-Roig, and the second author [@MMN], as well as Zanello and Zylinski [@ZZ], began explorations into the connection between the weak Lefschetz property and positive characteristic, and also posed several interesting questions.

In [@CN], the authors found a connection between certain families of level artinian monomial almost complete intersections and lozenge tilings of hexagons; independently, Li and Zanello [@LZ] found a similar connection for artinian monomial complete intersections (see also Corollary \[cor:ci\]). However, both were without combinatorial bijection until one was found by Chen, Guo, Jin, and Liu [@CGJL]; Boyle, Migliore, and Zanello [@BMZ] have pushed this connection further. Brenner and Kaid [@BK-2010] also consider artinian monomial complete intersections in three variables with generators 