  Let $d(N )$ (resp. $p(N )$) be the number of summands in the determinant
(resp. permanent) of an $N\times N$ circulant matrix $A = (a_{ij} )$ given by
$a_{ij} = X_{i+j}$ where $i + j$ should be considered $\mod N$ . This short
note is devoted to prove that $d(N ) = p(N )$ if and only if $N$ is a prime
power. We then give an application to homogeneous monomial ideals failing the
Weak Lefschetz property.
