The Dittert conjecture, or Dittert–Hajek conjecture, is a mathematical hypothesis (in

combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...

) 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 Bruce Edward Hajek is a Professor in the Coordinated Science Laboratory, the head of the Department of Electrical and Computer Engineering, and the Leonard C. and Mary Lou Hoeft Chair in Engineering at the University of Illinois at Urbana-Champai ...

. Let

A =_/math> be a

square matrix In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are often ...

of order

n

with nonnegative entries and with

\sum_^n \left ( \sum_^n a_ \right ) = n

. Its

permanent Permanent may refer to: Art and entertainment * ''Permanent'' (film), a 2017 American film * ''Permanent'' (Joy Division album) * "Permanent" (song), by David Cook Other uses * Permanent (mathematics), a concept in linear algebra * Permanent (cy ...

is defined as

\operatorname(A)=\sum_\prod_^n a_

, where the sum extends over all elements

\sigma

of the symmetric group. The Dittert conjecture asserts that the function

\operatorname(A)

defined by

\prod_^n \left ( \sum_^n a_ \right ) + \prod_^n \left ( \sum_^n a_ \right ) - \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.

References

{{reflist Conjectures Combinatorics Inequalities