Dittert conjecture

The Dittert conjecture, or Dittert–Hajek conjecture, is a mathematical hypothesis in combinatorics concerning the maximum achieved by a particular function ${\displaystyle \phi }$ of matrices with real, nonnegative entries satisfying a summation condition. The conjecture is due to Eric Dittert and (independently) Bruce Hajek.^[1][2][3][4]

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

The Dittert conjecture asserts that the function ${\displaystyle \operatorname {\phi } (A)}$ 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 ${\displaystyle A=(1/n)J_{n}}$, where ${\displaystyle J_{n}}$ is defined to be the square matrix of order ${\displaystyle n}$ with all entries equal to 1.^[1][2]