MONOTONICTTY CONJECTURE ON PERMANENTS OF DOUBLY STOCHASTIC MATRICES i

A stronger version of the van der Waerden permanent conjecture asserts that if J„ denotes the n X n matrix all of whose entries are \/n and A is any fixed matrix on the boundary of the set ofnxn doubly stochastic matrices, then per(A/4 + (1 — \)Jn) as a function of X is nondecreasing in the interval [0, 1]. In this paper, we elucidate the relation of this assertion to some other conjectures known to be stronger than van der Waerden's. We also show that this assertion is true when n = 3 and in the case when, up to permutations of rows and columns, either (i) A = J, ffi /,, 0 < s, t, j + (=nor (ii) A — [yr £] where OisanjXs zero matrix, Y is s X t with all entries equal to 1/r, and Z is t x t with all entries equal to (t s)/t2, 0<s<t,s + t = n.