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a b s t r a c tWe 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).

1, 3, 6, 10, 15, 21, 28, 27, 27, 28) is a level h-vector! This example answers negatively the open question as to whether all codimension 3 level h-vectors are unimodal.Moreover, using the same (simple) technique, we are able to construct level algebras of codimension 3 whose h-vectors have exactly N "maxima," for any positive integer N .These non-unimodal h-vectors, in particular, provide examples of codimension 3 level algebras not enjoying the Weak Lefschetz Property (WLP). Their existence was also an open problem before.In the second part of the paper we further investigate this fundamental property, and show that there even exist codimension 3 level algebras of type 3 without the WLP.

Abstract. In this short paper we establish a (non-trivial) lower bound on the degree two entry h 2 of a Gorenstein h-vector of any given socle degree e and any codimension r.In particular, when e = 4, that is, for Gorenstein h-vectors of the form h = (1, r, h 2 , r, 1), our lower bound allows us to prove a conjecture of Stanley on the order of magnitude of the minimum value, say f (r), that h 2 may assume. In fact, we show thatIn general, we wonder whether our lower bound is sharp for all integers e ≥ 4 and r ≥ 2.