A003432 -id:A003432 - cp's OEIS frontend

A215645 Depth for {+1,-1} maximal determinant matrices: minimal depth for which a proper submatrix is also a maximal determinant matrix.

1, 1, 1, 1, 1, 1, 1, 4, 1, 2, 3, 5, 6, 7, 8, 8, 1, 7, 10, 10, 10

Offset: 1

Richard P. Brent and Judy-anne Osborn, Aug 18 2012

The complementary depth m(A) of a maximal determinant {+1,-1} matrix of order n is the maximum m < n such that a maximal determinant matrix of order m occurs as a proper submatrix of A, or 0 if n = 1. The depth d(A) of A is d(A) := n - m(A). The depth d(n) is the minimum of d(A) over all maximal determinant matrices A of order n.

We calculated the first 21 terms of the sequence by an exhaustive computation of minors of known maximal determinant matrices as of August 2012.

			For n = 11 the depth is 3 because there is a maximal determinant matrix of order 11 that has a maximal determinant submatrix of order 8 = 11-3, but no larger proper maximal determinant submatrices. Note that only one of the three Hadamard equivalence classes of maximal determinant matrices of order 11 gives depth 3; the others give depth 4, but we take the minimum.

R. P. Brent, The Hadamard Maximal Determinant Problem
Richard P. Brent and Judy-anne H. Osborn, On minors of maximal determinant matrices, arXiv:1208.3819, 2012.

Cf. A003432, A003433, A215644.

A276410 Largest determinant of a (real) {0,1}-matrix of order n subject to the restriction that the corresponding 0,1 simplex is acute.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 9, 32, 56, 96

Offset: 0

N. J. A. Sloane, Sep 17 2016

Jan Brandts, A. Cihangir, Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group, arXiv preprint arXiv:1512.03044 [math.CO], 2015. See Fig. 1.

See A003432 for the usual version of this problem.

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

Original entry on oeis.org

0, 2, 2, 3, 4, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64

Offset: 2

Hugo Pfoertner, Mar 20 2018

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)).

The sharper bound is proved by Araujo, Balogh, and Wang in their article. See link. - Hugo Pfoertner, Nov 04 2020

			a(8) = 6 because no (0,1)-matrix with 2*8 ones with a greater determinant exists than
  ( 1 0 0 0 0 0 0 0 )
  ( 0 1 0 1 0 0 0 0 )
  ( 0 0 1 0 1 1 0 0 )
  ( 0 0 0 1 0 0 1 0 )
  ( 0 0 0 0 1 0 0 1 )
  ( 0 0 0 0 0 1 0 1 )
  ( 0 1 0 0 0 0 1 0 )
  ( 0 0 1 0 0 0 0 1 )

Igor Araujo, József Balogh, and Yuzhou Wang, Maximum determinant and permanent of sparse 0-1 matrices, arXiv:2011.01892 [math.CO], 3 Nov 2020.
Henning Bruhn and Dieter Rautenbach, Maximal determinants of combinatorial matrices, arXiv:1711.09935 [math.CO], 2017.
Mathoverflow, Are bounds known for the maximum determinant of a (0,1)-matrix of specified size and with a specifed number of 1s?, 2014-2018.
Yaroslav Shitov, On the determinant of a sparse 0-1 matrix, Linear Algebra and its Applications, Volume 554, 1 October 2018, Pages 49-50.
Markus Sigg, Gasper's determinant theorem, revisited, arXiv:1804.02897 [math.CO], 2018.

Cf. A003432, A017979, A085000.

A352348 Maximum determinant of n X n matrix composed of {-1, 0, 1} with pairwise orthogonal rows.

Original entry on oeis.org

1, 2, 2, 16, 16, 125, 128, 4096, 4096, 59049, 59049, 2985984, 2985984, 62748517

Offset: 1

Max Alekseyev, Mar 12 2022

a(n) >= a(m)*a(n-m) for any m < n.
a(n) <= A003433(n), a bound achieved if the orthogonality requirement is dropped.
If there exists an order n Hadamard matrix, then a(n) = A003433(n) = n^(n/2).
For n == 2 (mod 4), if there exists an order n conference matrix (cf. A000952), then a(n) = (n-1)^(n/2). In particular, a(18) = 118587876497.

Arun et al., Reference request: maximal determinant of matrices with pairwise orthogonal rows and entries in {1, 0, -1}, MathOverflow, 2022.
Wikipedia, Conference matrix.
Wikipedia, Hadamard matrix.

Cf. A000952, A003432, A003433.

a(11)-a(14) from Max Alekseyev, May 20 2023

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A215645 Depth for {+1,-1} maximal determinant matrices: minimal depth for which a proper submatrix is also a maximal determinant matrix.

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A276410 Largest determinant of a (real) {0,1}-matrix of order n subject to the restriction that the corresponding 0,1 simplex is acute.

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

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A352348 Maximum determinant of n X n matrix composed of {-1, 0, 1} with pairwise orthogonal rows.

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