  In this paper, we determine the maximum $h_{max}$ and the minimum $h_{min}$
of the Hilbert vectors of Perazzo algebras $A_F$, where $F$ is a Perazzo
polynomial of degree $d$ in $n+m+1$ variables. These algebras always fail the
Strong Lefschetz Property. We determine the integers $n,m,d$ such that
$h_{max}$ (resp. $h_{min}$) is unimodal, and we prove that $A_F$ always fails
the Weak Lefschetz Property if its Hilbert vector is maximum, while it
satisfies the Weak Lefschetz Property if it is minimum, unimodal, and satisfies
an additional mild condition. We determine the minimal free resolution of
Perazzo algebras associated to Perazzo threefolds in $\mathbb P^4$ with minimum
Hilbert vectors. Finally we pose some open problems in this context.
  Dedicated to Enrique Arrondo on the occasion of his $60^{th}$ birthday.
