  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).
