the depth of the problem by considering a larger class of algebras and relating the problem to [*a priori*]{} seemingly unrelated questions in combinatorics and algebraic geometry. This builds on the work of many authors (e.g., [@BK], [@CGJL], [@CN], [@LZ], and [@MMN]).

Throughout this work we consider in particular the question of how the weak Lefschetz property of a certain $K$-algebra $A$ depends on the characteristic of the field $K$. We begin by relating the algebra $A$ to two square integer matrices, $N$ and $Z$, where the entries of $N$ are binomial coefficients and $Z$ is a zero-one matrix. We show that $A$ has the weak Lefschetz property if and only if the determinant of either of these matrices does not vanish modulo the characteristic of $K$. Next, we establish that the determinant of $N$ enumerates signed lozenge tilings of a punctured hexagonal region and that the determinant of $Z$ enumerates signed perfect matchings of a bipartite graph associated to the same punctured hexagonal region. The relation to the weak Lefschetz property implies that both determinants have the same prime divisors; in fact, we show that their absolute values are the same by using combinatorial arguments. Finally, we show that 