Dittert conjecture explained

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.^{[1] [2]}

Let

A = [a _{ij} ] be a square matrix of order n with nonnegative entries and with $\backslash sum\_^n\; \backslash left\; (\backslash sum\_^n\; a\_\; \backslash right)\; =\; n$. Its permanent is defined as$$\backslash operatorname(A)=\backslash sum\_\backslash prod\_^n\; a\_,$$where the sum extends over all elements \sigma of the symmetric group.

The Dittert conjecture asserts that the function

\operatorname{\phi}(A) defined by $\backslash prod\_^n\; \backslash left\; (\backslash sum\_^n\; a\_\; \backslash right)\; +\; \backslash prod\_^n\; \backslash left\; (\backslash sum\_^n\; a\_\; \backslash right)\; -\; \backslash operatorname(A)$ 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.^{[1] [2]}

Notes and References

1. Book: Hogben, Leslie. Leslie Hogben . Handbook of Linear Algebra. 2nd. CRC Press. 2014. 43–8.
2. 436. 4. 15 February 2012. 791–801. Some results towards the Dittert conjecture on permanents. Linear Algebra and its Applications. Cheon, Gi-Sang. Wanless, Ian M.. 10.1016/j.laa.2010.08.041. free. 1885/28596. free.