Some Conjectures on Permanents of Doubly Stochastic Matrices

Let Omega(n) denote the set of all doubly stochastic matrices of order n. Foregger [3] raised a question whether per (tJ(n)+(1-t)A) <= per (A) holds for all A is an element of Omega(n) and 0 < t <= n/n-1, where J(n) is the n x n matrix with each entry equal to 1/n. But this inequality does not hold good for all matrices in general. In this paper, we consider the above inequality for subpermanents and we provide a sufficient condition for a matrix Lambda is an element of Omega(n) to satisfy the inequality sigma(k)(tJ(n)+(1-t)A) <= sigma(k)(A) for 0 <= t <= 1 and discuss the consequences of this inequality.