connections to show that the determinant of the zero-one matrix $Z_A$ and the binomial matrix $N_A$ are the same, up to sign (Theorem \[thm:det-Z-N\]). Moreover, in a special case of Kasteleyn’s theorem [@Ka] about enumerating perfect matchings, when the puncture has an even side-length, then the determinant and the permanent of $Z_A$ are also the same, up to sign (Corollary \[cor:det-Z-per-Z\]).

In Section \[sec:det\] we prove first that the determinant of $N_A$ is non-zero when the puncture is of even side-length (Theorem \[thm:M-even\]), thus establishing the weak Lefschetz property in many new cases. We then find closed formulae for the determinants when the puncture is trivial (Proposition \[pro:M-zero\]), when any one side of the hexagonal region has length zero (Proposition \[pro:C-zero\]), when a vertex of the puncture touches one of the sides of the region (Proposition \[pro:C-maximal\]), and when a side of the puncture touches one of the sides of the region (Proposition \[pro:gamma-zero\]). We close with a complete description of when the region is symmetric. In particular, we show that when certain parity conditions hold the determinant is zero (Proposition \[pro:symmetry-zero\]) and we provide a conjecture for a closed formula of the determinant when the same parity conditions fail 