The Dittert conjecture, or Dittert–Hajek conjecture, is a mathematical hypothesis in combinatorics concerning the maximum achieved by a particular function ϕ {\displaystyle \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 i j ] {\displaystyle A=[a_{ij}]} be a square matrix of order n {\displaystyle n} with nonnegative entries and with ∑ i = 1 n ( ∑ j = 1 n a i j ) = n {\textstyle \sum _{i=1}^{n}\left(\sum _{j=1}^{n}a_{ij}\right)=n} . Its permanent is defined as per ⁡ ( A ) = ∑ σ ∈ S n ∏ i = 1 n a i , σ ( i ) , {\displaystyle \operatorname {per} (A)=\sum _{\sigma \in S_{n}}\prod _{i=1}^{n}a_{i,\sigma (i)},} where the sum extends over all elements σ {\displaystyle \sigma } of the symmetric group.

The Dittert conjecture asserts that the function ϕ ⁡ ( A ) {\displaystyle \operatorname {\phi } (A)} defined by ∏ i = 1 n ( ∑ j = 1 n a i j ) + ∏ j = 1 n ( ∑ i = 1 n a i j ) − per ⁡ ( A ) {\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 {\displaystyle A=(1/n)J_{n}} , where J n {\displaystyle J_{n}} is defined to be the square matrix of order n {\displaystyle n} with all entries equal to 1.