The Number of Terms in the Permanent and the Determinant of a Generic Circulant Matrix

Let A = (aij) be the generic n×n circulant matrix given by aij = xi+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). Introduction A square matrix is said to be a circulant matrix if its rows are successive cyclic permutations of the first row. Thus, the matrix A = (aij) with aij = xi+j , subscripts on x being interpreted mod n, is a generic circulant matrix. If we expand det(A), we obtain a polynomial in the xi. We define d(n) to be the number of terms in this polynomial after like terms have been combined. Similarly, we define p(n) to be the number of terms in per(A), the permanent of A. The function p(n) was studied in Brualdi and Newman [1], where it was pointed out that the main result of Hall [3] shows that p(n) coincides with the number of solutions to y1 + 2y2 + · · ·+ nyn ≡ 0 (mod n) y1 + · · ·+ yn = n (1) in non-negative integers. Using this formulation, they showed by a generating function argument that