The weak Lefschetz property for monomial ideals of small type

In this work a combinatorial approach towards the weak Lefschetz property is developed that relates this property to enumerations of signed perfect matchings as well as to enumerations of signed families of non-intersecting lattice paths in certain triangular regions. This connection is used to study Artinian quotients by monomial ideals of a three-dimensional polynomial ring. Extending a main result in the recent memoir [Boij, Migliore, Mir\'o-Roig, Nagel, Zanello], we completely classify the quotients of type two that have the weak Lefschetz property in characteristic zero. We also derive results in positive characteristic for quotients whose type is at most two.