  We introduce a natural correspondence between artinian monomial almost
complete intersections in three variables and punctured hexagonal regions. We
use this correspondence to investigate the algebras for the presence of the
weak Lefschetz property. In particular, we relate the field characteristics in
which such an algebra fails to have the weak Lefschetz property to the prime
divisors of the enumeration of signed lozenge tilings of the associated
punctured hexagonal region. On the one side this allows us to establish the
weak Lefschetz property in many new cases.
  For numerous classes of punctured hexagonal regions we find closed formulae
for the enumerations of signed lozenge tilings, and thus the field
characteristics in which the associated algebras fail to the have the weak
Lefschetz property. Further, we offer a conjecture for a closed formula for the
enumerations of signed lozenge tilings of symmetric punctured hexagonal
regions. These formulae are exploited to lend further evidence to a conjecture
by Migliore, Mir\'o-Roig, and the second author that classifies the {\em level}
artinian monomial almost complete intersections in three variables that have
the weak Lefschetz property in characteristic zero. Moreover, the formulae are
used to generate families of algebras which never, or always, have the weak
Lefschetz property, regardless of field characteristic. Finally, we determine
(in one case, depending on the presence of the weak Lefschetz property) the
splitting type of the syzygy bundle of an artinian monomial almost complete
intersection in three variables, when the characteristic of the base field is
zero.
  Our results convey an intriguing interplay between problems in algebra,
combinatorics, and algebraic geometry, which raises new questions and deserves
further investigation.
