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

الأصل · EN

Let A=(a(ij)) be the generic n by n circulant matrix given by a(ij)=x(i+j), with subscripts on x interpreted mod n. Define d(n) (resp. p(n)) to be the number of terms in the determinant (resp. permanent) of A. The function p(n) is well-known and has several combinatorial interpretations. The function d(n), on the other hand, has not been studied previously. We show that when n is a prime power, d(n)=p(n). The proof uses symmetric functions.