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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 perfectmatchings, and to families of non-intersecting lattice paths. In this work weconsider more general regions, called triangular regions, and establish signedversions of the latter three bijections. Indeed, we use perfect matchings andfamilies of non-intersecting lattice paths to define two signs of a lozengetiling. A combinatorial argument involving a new method, called resolution of apuncture, then shows that the signs are in fact equivalent. This provides inparticular two different determinantal enumerations of these families. Theseresults are then applied to study the weak Lefschetz property of Artinianquotients by monomial ideals of a three-dimensional polynomial ring. Weestablish sufficient conditions guaranteeing the weak Lefschetz property aswell as the semistability of the syzygy bundle of the ideal, classify the typetwo algebras with the weak Lefschetz property, and study monomial almostcomplete intersections in depth. Furthermore, we develop a general method thatoften associates to an algebra that fails the weak Lefschetz property a toricsurface that satisfies a Laplace equation. We also present examples of toricvarieties that satisfy arbitrarily many Laplace equations. Our combinatorialmethods allow us to address the dependence on the characteristic of the basefield for many of our results.

Enumerations of lozenge tilings, lattice paths, and perfect matchings and the weak Lefschetz property