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abstract: |
    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. On the other side we can determine some of the prime divisors of the enumerations by means of an algebraic argument.

    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, Miró-Roig, and the second author that classifies the [*level*]{} artinian monomial almost complete intersections in three variables that have 