cp's OEIS Frontend

A301370 Maximum determinant of an n X n (0,1)-matrix that has exactly 2*n ones.

%I A301370 #27 Nov 04 2020 10:59:09
%S A301370 0,2,2,3,4,4,6,8,9,12,16,18,24,32,36,48,64
%N A301370 Maximum determinant of an n X n (0,1)-matrix that has exactly 2*n ones.
%C A301370 A proved upper bound is abs(a(n)) <= 6^(n/6), provided by Bruhn and Rautenbach. A conjectured sharper bound is abs(a(n)) <= 2^(n/3), provided by the same authors. For n=3*k, the bound is achieved by diagonally concatenating blocks ((1 1 0)(0 1 1)(1 0 1)).
%C A301370 The sharper bound is proved by Araujo, Balogh, and Wang in their article. See link. - _Hugo Pfoertner_, Nov 04 2020
%H A301370 Igor Araujo, József Balogh, and Yuzhou Wang, <a href="https://arxiv.org/abs/2011.01892">Maximum determinant and permanent of sparse 0-1 matrices</a>, arXiv:2011.01892 [math.CO], 3 Nov 2020.
%H A301370 Henning Bruhn and Dieter Rautenbach, <a href="https://arxiv.org/abs/1711.09935">Maximal determinants of combinatorial matrices</a>, arXiv:1711.09935 [math.CO], 2017.
%H A301370 Mathoverflow, <a href="https://mathoverflow.net/questions/180604/are-bounds-known-for-the-maximum-determinant-of-a-0-1-matrix-of-specified-size">Are bounds known for the maximum determinant of a (0,1)-matrix of specified size and with a specifed number of 1s?</a>, 2014-2018.
%H A301370 Yaroslav Shitov, <a href="https://doi.org/10.1016/j.laa.2018.05.019">On the determinant of a sparse 0-1 matrix</a>, Linear Algebra and its Applications, Volume 554, 1 October 2018, Pages 49-50.
%H A301370 Markus Sigg, <a href="https://arxiv.org/abs/1804.02897">Gasper's determinant theorem, revisited</a>, arXiv:1804.02897 [math.CO], 2018.
%e A301370 a(8) = 6 because no (0,1)-matrix with 2*8 ones with a greater determinant exists than
%e A301370   ( 1 0 0 0 0 0 0 0 )
%e A301370   ( 0 1 0 1 0 0 0 0 )
%e A301370   ( 0 0 1 0 1 1 0 0 )
%e A301370   ( 0 0 0 1 0 0 1 0 )
%e A301370   ( 0 0 0 0 1 0 0 1 )
%e A301370   ( 0 0 0 0 0 1 0 1 )
%e A301370   ( 0 1 0 0 0 0 1 0 )
%e A301370   ( 0 0 1 0 0 0 0 1 )
%Y A301370 Cf. A003432, A017979, A085000.
%K A301370 nonn,hard,more
%O A301370 2,2
%A A301370 _Hugo Pfoertner_, Mar 20 2018