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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 $\frac{1}{n}$. Thisconjecture was proved partially for $n \leq 5$. \\ \indent Let $K_n$ denote theset 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 $X\in K_n$ withrow sum vector $(r_1,r_2,...r_n)$ and column sum vector $(c_1,c_2,...c_n)$. Amatrix $A\in K_n$ 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 $n=2$ andHwang proved it for $n=3$. \\ \indent In this paper, we prove the Lih and Wangconjecture for $n=6$ and Dittert conjecture for $n=4$.

Lih Wang and Dittert Conjectures on Permanents