# Lih Wang and Dittert Conjectures on Permanents Source: [Lih Wang and Dittert Conjectures on Permanents]() Collection index: [llms.txt]() Data updated: 2025-10-25T16:54:33.445Z Authors: Divya.K.U, K. Somasundaram Published: 2023-12-01T09:58:02.000Z ## 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 $n=4$. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. ## Original sources and implementation [Original paper]() [PDF]()