On the Coefficients of the Permanent and the Determinant of a Circulant Matrix. Applications

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 $\mathrm{m}\mathrm{o}\mathrm{d}N$ . 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.