Lih Wang and Dittert Conjectures on Permanents

Abstract

Let $\Omega_n$ denote the set of all doubly stochastic matrices of order $n$. Lih and Wang conjectured that for $n\geq3$, per$(tJ_n+(1-t)A)\leq t $per$J_n+(1-t)$per$A$, for all $A\in\Omega_n$ and all $t \in [0.5,1]$, where $J_n$ is the $n \times n$ matrix with each entry equal to $\frac{1}{n}$. This conjecture was proved partially for $n \leq 5$. \\ \indent Let $K_n$ denote the set of non-negative $n\times n$ matrices whose elements have sum $n$. Let $\phi$ be a real valued function defined on $K_n$ by $\phi(X)=\prod_{i=1}^{n}r_i+\prod_{j=1}^{n}c_j$ - per$X$ for $X\in K_n$ with row sum vector $(r_1,r_2,...r_n)$ and column sum vector $(c_1,c_2,...c_n)$. A matrix $A\in K_n$ is called a $\phi$-maximizing matrix if $\phi(A)\geq \phi(X)$ for all $X\in K_n$. Dittert conjectured that $J_n$ is the unique $\phi$-maximizing matrix on $K_n$. Sinkhorn proved the conjecture for $n=2$ and Hwang proved it for $n=3$. \\ \indent In this paper, we prove the Lih and Wang conjecture for $n=6$ and Dittert conjecture for $n=4$.