The Dittert conjecture, or Dittert–Hajek conjecture, is a mathematical hypothesis (in combinatorics) concerning the maximum achieved by a particular function ϕ {\displaystyle \phi } \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 n with nonnegative entries and with \( {\displaystyle \sum _{i=1}^{n}\left(\sum _{j=1}^{n}a_{ij}\right)=n} \). Its permanent is defined as per \( {\displaystyle \operatorname {per} (A)=\sum _{\sigma \in S_{n}}\prod _{i=1}^{n}a_{i,\sigma (i)}} \), where the sum extends over all elements \( \sigma \) of the symmetric group.
The Dittert conjecture asserts that the function \( {\displaystyle \operatorname {\phi } (A)} \) defined by \( {\displaystyle \prod _{i=1}^{n}\left(\sum _{j=1}^{n}a_{ij}\right)+\prod _{j=1}^{n}\left(\sum _{i=1}^{n}a_{ij}\right)-\operatorname {per} (A)} \) is (uniquely) maximized when \( {\displaystyle A=(1/n)J_{n}} \), where \( J_n \) is defined to be the square matrix of order } n with all entries equal to 1.[1][2]
References
Hogben, Leslie, ed. (2014). Handbook of Linear Algebra (2nd ed.). CRC Press. pp. 43–8.
Cheon, Gi-Sang; Wanless, Ian M. (15 February 2012). "Some results towards the Dittert conjecture on permanents". Linear Algebra and its Applications. 436 (4): 791–801. doi:10.1016/j.laa.2010.08.041.
Eric R. Dittert at the Mathematics Genealogy Project
Bruce Edward Hajek at the Mathematics Genealogy Project
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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