Dittert conjecture

The Dittert conjecture, or Dittert–Hajek conjecture, is a mathematical hypothesis in combinatorics concerning the maximum achieved by a particular function $$\phi$$ of matrices with real, nonnegative entries satisfying a summation condition. The conjecture is due to Eric Dittert and (independently) Bruce Hajek.

Let $$A = [a_{ij}]$$ be a square matrix of order $$n$$ with nonnegative entries and with $ \sum_{i=1}^n \left ( \sum_{j=1}^n a_{ij} \right ) = n $. Its permanent is defined as $$ \operatorname{per}(A)=\sum_{\sigma\in S_n}\prod_{i=1}^n a_{i,\sigma(i)}, $$ where the sum extends over all elements $$\sigma$$ of the symmetric group.

The Dittert conjecture asserts that the function $$\operatorname{\phi}(A)$$ defined by $\prod_{i=1}^n \left ( \sum_{j=1}^n a_{ij} \right ) + \prod_{j=1}^n \left ( \sum_{i=1}^n a_{ij} \right ) - \operatorname{per}(A)$ is (uniquely) maximized when $$A = (1/n) J_n$$, where $$J_n$$ is defined to be the square matrix of order $$n$$ with all entries equal to 1.