Ryser, Glynn, and the discrete Fourier transform: orthogonal schemes for the permanent

Abstract

The permanent of an $n \times n$ matrix is the coefficient of the fully mixed monomial $x_0 \cdots x_{n-1}$ in the product of its row forms $\prod_i\bigl(\sum_j b_{ij} x_j\bigr)$, and the classical exact algorithms of Ryser and Glynn compute it by summing $2^{n-1}$ evaluations of that product over the Boolean cube. We recast this as a single principle -- an evaluation scheme together with an orthogonality criterion that decides, in one line, whether its weighted sum equals the permanent -- and show that Ryser's formula, Glynn's formula, and a discrete Fourier transform are three instances of it: the $0$--$1$ and $\pm 1$ Hadamard systems, and a single cyclic transform of length $N$. The DFT scheme evaluates $\mathrm{per}\,B$ as one coefficient of a univariate polynomial modulo $x^N - 1$; it is exact exactly when the exponent set is valid mod $N$, runs in the same $\Theta(2^n n)$ operations as the classical formulas, and over a finite field is a number-theoretic transform that returns the exact integer permanent by the Chinese remainder theorem. Where the Hadamard schemes are orthogonal for free, the cyclic scheme trades this for an arithmetic condition on the exponents; the smallest length at which it can be met is $N = 2^n - 2^{\lfloor \log_2 n \rfloor}$, established in the companion paper arXiv:2607.08366. All three are character sums over a finite abelian group, and the criterion holds precisely for such groups.