the weak Lefschetz property in positive characteristic $p$ if and only if $p$ is not a prime divisor of the determinant of $Z_A$ (Proposition \[pro:wlp-zero-one\]). We also describe a matrix $N_A$ with binomial entries that also has the analogous property (Proposition \[pro:wlp-binom\]). Moreover, we demonstrate that a rather simple algebraic argument can be used to determine some of the prime divisors of the determinants for both $Z_A$ and $N_A$ (Proposition \[pro:wlp-p\]).

In Section \[sec:ph\] we organise the monomials generating the peak homogeneous components of such an algebra in a plane. It turns out that the monomials fill a punctured hexagon (Theorem \[thm:interlace-amaci\]). Using the well-known bijection between lozenge tilings and non-intersecting lattice paths, and the Lindström-Gessel-Viennot theorem ([@GV], [@GV-1989], [@Li]) on non-intersecting lattice paths, we show that the determinant of the binomial matrix $N_A$ is the enumeration of the signed lozenge tilings of the punctured hexagon, up to sign (Theorem \[thm:nilp-matrix\]). Furthermore, using another well-known bijection between lozenge tilings and perfect matchings (see, e.g., [@Ku]), we argue that the determinant of the zero-one matrix $Z_A$ is an enumeration of the signed perfect matchings of the associated bipartite graph, up to sign (Theorem \[thm:bip-matrix\]).

In Section \[sec:signs\] we use the aforementioned 