The number of terms in the permanent and the determinant of a generic circulant matrix (Q703037)

Let \(A =(a_{ij})\) with \(a_{ij}=x_{i+j}\), \(j=1,2,\dots,n\), subscripts on \(x\) being interpreted \(\bmod n\), be a generic circulant matrix, \(d(n)\) be the number of terms in \(\text{det}(A)\) after like terms have been combined, and \(p(n)\) be the number of terms in \(\text{per}(A)\), the permanent of \(A\). Using the theory of symmetric functions, the author proves that if \(n\) is a prime power, \(d(n)=p(n)\).