Permanents of convex combinations of doubly stochastic matrices
Thomas H. Foregger · Linear and Multilinear Algebra · 1988
Let S be a 4 × 4 doubly stochastic matrix and t 0 < t ⩾ 4/3, where t 0 is the unique real root of 106t 3 − 418t 2 + 465t − 100. We prove that per(tJ 4 + (1 − t)S) ⩽ per(S) with equality if and only if S = J 4. This confirms for n = 4 a conjecture of K. Lih and E. T. H. Wang that for . We also show that another conjecture of K. Lih and E. T. H. Wang, per(tjn + (1 - t) S) ⩽ t per(Jn ) + (1 − t) per(S) for t ε [1/2, 1] is true for n = 4 and t ε [t 2, 1], where t 2 is the unique real root of 106t 3 − 418t 2 + 465t − 153. Note that t 2 is about 0.6216986477375. Finally, we exhibit a set of 4 × 4 doubly stochastic matrices for which per(tJ 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.