Permanents of doubly stochastic matrices with diagonal submatrices of fixed order. (Q1415289)

For positive integers \(r\) and \(n\) with \(n>r\), let \(D_{r, n}=\left( \begin{smallmatrix} 0_r & J\\ J & I_n \end{smallmatrix} \right)\), where the \(J\)s are matrices of \(1\)s of suitable sizes, and let \(\Omega(D_{r,n})\) denote the face of the polytope \(\Omega_{r+n}\) consisting of all \((r+n)\)-square doubly stochastic matrices \(A\leq D_{r, n}\). The authors prove that the barycenter of \(\Omega(D_{r,n})\) is a local minimum point for the permanent function over \(\Omega(D_{r, n})\), and that if a matrix \(A=(a_{ij})\) is a minimum point for the permanent function over \(\Omega(D_{r, n})\) such that \(a_{ij}\leq \frac {n-r}{n-1}\), \(i=r+1,\dots, r+n\), then \(A\) is the barycenter of \(\Omega(D_{r,n})\).