The Kopotun Subpermanent Transform Inequality

We prove the subpermanent transform inequality conjectured by Kopotun in 1996: for every doubly stochastic matrix A of order n ≥ 2 and every 2 ≤ k ≤ n, σk(A) ≥ σk((nJn + A)/(n + 1)). Here σk is the sum of all subpermanents of order k, and Jn has every entry 1/n. Equality holds exactly at A = Jn.