Monomial complete intersections, the weak Lefschetz property and plane partitions

We characterize the monomial complete intersections in three variables satisfying the Weak Lefschetz Property (WLP), as a function of the characteristic of the base field. Our result presents a surprising, and still combinatorially obscure, connection with the enumeration of plane partitions. It turns out that the rational primes pp dividing the number, M(a,b,c)M(a,b,c), of plane partitions contained inside an arbitrary box of given sides a,b,ca,b,c are precisely those for which a suitable monomial complete intersection (explicitly constructed as a bijective function of a,b,ca,b,c) fails to have the WLP in characteristic pp. We wonder how powerful can be this connection between combinatorial commutative algebra and partition theory. We present a first result in this direction, by deducing, using our algebraic techniques for the WLP, some explicit information on the rational primes dividing M(a,b,c)M(a,b,c).