  The purpose of this note is to characterize the finite Hilbert functions
which force all of their artinian algebras to enjoy the Weak Lefschetz Property
(WLP). Curiously, they turn out to be exactly those (characterized by Wiebe in
$[Wi]$) whose Gotzmann ideals have the WLP.
  This implies that, if a Gotzmann ideal has the WLP, then all algebras with
the same Hilbert function (and hence lower Betti numbers) have the WLP as well.
However, we will answer in the negative, even in the case of level algebras,
the most natural question that one might ask after reading the previous
sentence: If $A$ is an artinian algebra enjoying the WLP, do all artinian
algebras with the same Hilbert function as $A$ and Betti numbers lower than
those of $A$ have the WLP as well?
  Also, as a consequence of our result, we have another (simpler) proof of the
fact that all codimension 2 algebras enjoy the WLP (this fact was first proven
in $[HMNW]$, where it was shown that even the Strong Lefschetz Property holds).
