(Conjecture \[con:symmetry\]).

In Section \[sec:central\] we explore two different ways to centralise the puncture. We call the puncture [*axis-central*]{} when it is central along each of the three axes, independently. Using very involved computations, Ciucu, Eisenkölbl, Krattenthaler, and Zare [@CEKZ] found closed formulae for the enumerations and signed enumerations of regions with an axis-central puncture; therein axis-central is called simply “central”. We use these closed formulae to describe the permanents of the zero-one matrices $Z_A$ (Corollary \[cor:axis-central-perm-Z\]) and the determinants of both matrices (Corollary \[cor:axis-central-det-Z\]), $Z_A$ and $N_A$, when the puncture is axis-central. We call the puncture [*gravity-central*]{} when its vertices are equidistant from the sides of the containing hexagon; this condition is equivalent to the associated algebra being level, that is, its socle is concentrated in one degree. Using this observation we provide further evidence for a conjecture by Migliore, Miró-Roig, and the second author [@MMN] about the presence of the weak Lefschetz property for level artinian monomial almost complete intersections in characteristic zero (Proposition \[pro:level-wlp\]).

In Section \[sec:interesting\] we describe a method, for any positive integer $n$, to generate a subfamily of algebras whose associated matrices have determinant $n$ (Proposition \[pro:det-n\]). From this we generate a subfamily of 