Monotonicity conjecture on permanents of doubly stochastic matrices

Abstract:

A stronger version of the van der Waerden permanent conjecture asserts that if ${J_n}$ denotes the $n \times n$ matrix all of whose entries are $1/n$ and $A$ is any fixed matrix on the boundary of the set of $n \times n$ doubly stochastic matrices, then ${\text {per}}(\lambda A + (1 - \lambda ){J_n})$ as a function of $\lambda$ 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) $A = {J_s} \oplus {J_t}$, $0 < s$, $t$, $s + t = n$ or (ii) $A = \left [\begin {smallmatrix} 0 & Y \\ Y^T & Z\end {smallmatrix} \right ]$ where 0 is an $s \times s$ zero matrix, $Y$ is $s \times t$ with all entries equal to $1/t$, and $Z$ is $t \times t$ with all entries equal to $(t - s)/{t^2}$, $0 < s \leqslant t$, $s + t = n$.