An exact algebraic determination and a sharp exceptional-spectrum formula

For every order $n$ and every nonnegative spectrum
$\lambda = (\lambda_1, \dots, \lambda_n)$, the dossier gives an exact finite
algebraic prescription for the maximum $M_n(\lambda)$ of
$\operatorname{per}(U^* \operatorname{diag}(\lambda) U)$ over unitary $U$. A
maximizer is a critical point of the permanent on the similarity orbit, where
$A$ commutes with its permanental adjoint; Gröbner elimination of these
equations, with the spectrum imposed before elimination, yields a univariate
polynomial whose real roots in $[0, \max_i \lambda_i^n]$ are the candidate
values. The Haar moments $\int \operatorname{per}(U^* \operatorname{diag}(\lambda) U)^p\,dU$,
evaluated exactly by Schur–Weyl duality as weighted sums of normalized Schur
polynomials, select the maximum among these roots at an explicitly bounded
order $p$. The prescription is not a closed-form formula. For spectra with one
exceptional eigenvalue, $(\alpha, \beta, \dots, \beta)$, a separate theorem
shows that the maximum is one of two explicit candidates and determines the
transition between them and all maximizers.