Signed lozenge tilings

Some Algebras with the Weak Lefschetz Property

“…Now we briefly review a connection between monomial ideals and triangular regions; for a more thorough discussion see [5].…”

“…The authors have developed a combinatorial approach towards deciding the presence of the weak Lefschetz property for monomial algebras in three variables in [5,7]. It relies on a study of lozenge tilings of so-called triangular regions.…”

Syzygy bundles and the weak Lefschetz property of almost complete intersections

“…Besides introducing notation, we recall needed facts from the combinatorial approach to Lefschetz properties developed in [9,11]. We also establish a new criterion for tileability by lozenges.…”

“…Any subregion T ⊂ T d can be associated to a bipartite planar graph G that is an induced subgraph of a honeycomb graph (see [9]). We are interested in the bi-adjacency matrix Z(T ) of G. This is a zero-one matrix whose determinant enumerates signed lozenge tilings (see [9,Theorem 3.5]). If T = T d (I) for some monomial ideal I, then Z(T ) admits an alternative description.…”

“…We focus on the case, where I has four minimal generators, extending previous work in [4,8,24]. To this end we use a combinatorial approach developed in [9,11] that involves lozenge tilings, perfect matchings, and families of non-intersecting lattice paths. Some of our results have already been used in [26].…”