  We study the weak Lefschetz property and the Hilbert function of level
Artinian monomial almost complete intersections in three variables. Several
such families are shown to have the weak Lefschetz property if the
characteristic of the base field is zero or greater than the maximal degree of
any minimal generator of the ideal. Two of the families have an interesting
relation to tilings of hexagons by lozenges. This lends further evidence to a
conjecture by Migliore, Miro-Roig, and the second author. Finally, using our
results about the weak Lefschetz property, we show that the Hilbert function of
each level Artinian monomial almost complete intersection in three variables is
peaked strictly unimodal.
