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 p dividing the number, M ( a , b , c ) , of plane partitions contained inside an arbitrary box of given sides a , b , c are precisely those for which a suitable monomial complete intersection (explicitly constructed as a bijective function of a , b , c ) fails to have the WLP in characteristic p . 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 ) .