Let d(N) (resp. p(N)) be the number of summands in the determinant (resp. permanent) of an N x 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.
