On a conjecture of E. Dittert
Let \(K_ n\) denote the set of \(n\times n\) nonnegative matrices whose entries have sum n. For an \(n\times n\) matrix \(X=(x_{ij})\in K_ n\), put \[ \phi (x)=\prod^{n}_{i=1}\sum^{n}_{j=1}x_{ij}+\prod^{n}_{j=1}\sum\;sp{n}_{i=1}x_{ij}-per X \] and for each i, j, put \(\phi_{ij}(x)=\prod_{k\neq i}r_ k+\prod_{k\neq j}c_ k-per X(i| j).\) Here \(r_ k\) is the kth row sum and \(c_ k\) is the kth column sum of X and X(i\(| j)\) is the (n-1)\(\times (n-1)\) matrix obtained by deleting the ith row and jth column of X, \(n>1\). If \(n=1\), \(X=(1)\) and \(\phi (X)=\phi_{11}(X)=1.\) A matrix \(A\in K_ n\) is \(\phi\)-maximizing if \(\phi (A)\geq \phi (X)\) for all \(X\in K_ n\). E. Dittert conjectured that if \(A\in K_ n\) is \(\phi\)-maximizing then \(A=J_ n\), the \(n\times n\) matrix whose entries all equal \(n^{-1}\). The author proves that the conjecture is true for \(n=3\) and shows that for any n if \(A\in K_ n\) is \(\phi\)-maximizing, then \(\phi_{ij}(A)=\phi (A)\) whenever \(\alpha_{ij}>0\) and \(\phi_{ij}(A)\leq \phi (A)\) whenever \(\alpha_{ij}=0\).
scientific article; zbMATH DE number 4021362