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

1,8

Comments

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.

Links

Table of n, a(n) for n=1..21.

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.

Example

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.

Crossrefs

Cf. A003432, A003433, A215644.

Sequence in context: A212497 A072046 A123609 * A075617 A055182 A298919

Adjacent sequences: A215642 A215643 A215644 * A215646 A215647 A215648

Keyword

nonn,hard,more

Author

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

Status

approved