Permanents of convex combinations of doubly stochastic matrices.

Let S be a 4 x 4 doubly stochastic matrix and t sub o =4/3, where t sub o is the unique real root of 106t sup 3 - 418t sup 2 + 465t - 100. We prove that per (tJ sub 4 +(1-t)S)=per (S) with equality if and only if S = J sub 4. This confirms for n=4 a conjecture of K. Lih and E.T.H. Wang that for S epsilon OMEGA sub n per (J sub n +S over 2) = per (S). We also show that another conjecture of K. Lih and E.T.H. Wang, per (tJ sub n +(1-t)S) =t per (J sub n)+(1-t)per (S) for t epsilon [1/2,1] is true for n=4 and t epsilon (t sub 2,1), where t sub 2 is the unique real root of 106t sup 3 -418t sup 2 -465t -153. Note that t sub 2 is about .6216986477375. Finally, we exhibit a set of 4 x 4 doubly stochastic matrices for which per (tJ sub 4 +(1-t)A) is a strictly decreasing function of t on (0,4/3) whenever A is a member of the set. This answers a question raised by H. Minc.