A003433 -id:A003433 - 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.

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

A385034 Absolute value squared of the maximal determinant of a matrix of order n with entries in the third roots of unity.

Original entry on oeis.org

1, 1, 3, 27, 189, 1701, 46656, 606528, 8957952, 387420489, 7360989291, 154580775111, 8916100448256, 222902511206400

Offset: 0

Guillermo N. Ponasso, Jun 16 2025

The n-th term is equal to n^n if and only if a Butson-type Hadamard matrix over the third roots exist.

			With w=exp(2*Pi*i/3) and i the imaginary unit, a matrix corresponding to a(3) = 27 is
  [1, 1,   1;
   1, w,   w^2;
   1, w^2, w],
and a matrix corresponding to a(4) = 189 is
  [w, 1, 1, 1;
   1, w, 1, 1;
   1, 1, w, 1;
   1, 1, 1, w].

G. Nuñez Ponasso, Maximal determinants of matrices over the roots of unity, Linear Algebra and its Applications, 723 (2025), 201-243.

Cf. A000312, A003433.

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.

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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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A385034 Absolute value squared of the maximal determinant of a matrix of order n with entries in the third roots of unity.

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