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

Author

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

Keywords

• nonn
• hard
• more

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.

Examples

			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.
		

Links

• 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.

Crossrefs

Cf. A003432, A003433, A215644.