  MacMahon enumerated the plane partitions in an $a \times b \times c$ box.
These are in bijection to lozenge tilings of a hexagon, to certain perfect
matchings, and to families of non-intersecting lattice paths. In this work we
consider more general regions, called triangular regions, and establish signed
versions of the latter three bijections. Indeed, we use perfect matchings and
families of non-intersecting lattice paths to define two signs of a lozenge
tiling. A combinatorial argument involving a new method, called resolution of a
puncture, then shows that the signs are in fact equivalent. This provides in
particular two different determinantal enumerations of these families. These
results are then applied to study the weak Lefschetz property of Artinian
quotients by monomial ideals of a three-dimensional polynomial ring. We
establish sufficient conditions guaranteeing the weak Lefschetz property as
well as the semistability of the syzygy bundle of the ideal, classify the type
two algebras with the weak Lefschetz property, and study monomial almost
complete intersections in depth. Furthermore, we develop a general method that
often associates to an algebra that fails the weak Lefschetz property a toric
surface that satisfies a Laplace equation. We also present examples of toric
varieties that satisfy arbitrarily many Laplace equations. Our combinatorial
methods allow us to address the dependence on the characteristic of the base
field for many of our results.
