Permanents of doubly stochastic matrices with diagonal submatrices of fixed order

For positive integers r, n with n r+1, let where Js denote the matrices of 1s of suitable sizes, and let Ω(Dr,n) denote the face of the polytope Ωr+n consisting of all (r+n)-square doubly stochastic matrices A such that A Dr,n. In this paper, the following are proved: (1)The barycenter of Ω(Dr,n) is a local minimum point for the permanent function over Ω(Dr,n).(2)If A=[aij] is a minimum point for the permanent function over Ω(Dr,n) such that aii (n−r)/(n−1), i=r+1,…,r+n, then A is the barycenter of Ω(Dr,n).