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 ${\textstyle \sum _{i=1}^{n}\left(\sum _{j=1}^{n}a_{ij}\right)=n}$. 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 ${\textstyle \prod _{i=1}^{n}\left(\sum _{j=1}^{n}a_{ij}\right)+\prod _{j=1}^{n}\left(\sum _{i=1}^{n}a_{ij}\right)-\operatorname {per} (A)}$ 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.