This paper presents a rare phenomenon—a nontrivial class of matrices (one of each size n) for which the permanent can be evaluated explicitly by an elegant formula. Let C denote the n-square matrix whose (i,j) entry is cij = cot[(2(j – i) + 1)p/ (2n)] and let J denote the n-square matrix of 1's. Then per(xJ + C) = n!(xn + n/2xn –- 2 + n(n – 2)/ (2·4)xn – 4 + (n(n – 2)(n – 4))/ (2 ·4 ·6) xn – 6 + ... ). To prove this formula a new method of evaluating permanents is developed, along with the standard methods due to Ryser and to Minc, from the unified point of view of Möbius inversion. The proof techniques then include trigonometric formulas involving roots of unity, counting combinatorial configurations of balls in cells in boxes, and a sign-reversing involution.
An identity of R. F. Scott, stated without proof in 1881, follows as a corollary. Proofs of Scott's identity did not appear until a hundred years later and it is something of a mystery how Scott obtained his identity. However, I present an argument that Scott himself was likely in possession of a proof, and the real mystery is why he did not publish it.
Last modified February 19, 2005