MONOTONICITY OF PERMANENTS OF CERTAIN DOUBLY STOCHASTIC MATRICES

Let pk(A), k = 1, , n, denote the sum of the permanents of all & X & submatrices of the n X n matrix A. We prove that where I n and P n are respectively the n x n identity matrix and the n X n permutation matrix with l's in positions (1, 2), (2, 3), , (n- 1, n), (n, 1). Using (*), we prove that for n Ξ> 3 and A = (J n + PJ/2, the functions are strictly monotonic increasing in the interval 0 ̂ θ:g 1