A stronger form of the Egorychev-Falikman theorem on permanents (Q762583)

Let A be an \(n\times n\) doubly stochastic matrix with columns \(a_ 1,...,a_ n\). Let \(A^{ij}\) be the matrix obtained from A by replacing each of the ith and the jth columns by \((a_ i+a_ j)\). Then at least one of the following conditions hold: (i) there exist i, j so that per \(A^{ij}<per A\), or (ii) if \(a_{ij}>0\) and \(a_{ik}>0\) then per \(A^{jk}\leq per A\). Furthermore if either A is partly decomposable or if \(A>0\) and some entry in A is not \(n^{-1}\), then (i) holds.