A347472 Maximum number of nonzero entries allowed in an n X n matrix to ensure there is a 2 X 2 zero submatrix.

0, 2, 6, 12, 19, 27, 39, 51, 65, 81, 98, 116, 139, 163, 188, 214, 242, 272, 303, 335, 375, 413, 453

Offset: 2

Author

M. F. Hasler, Sep 28 2021

Keywords

• nonn
• hard
• more

Comments

Related to Zarankiewicz's problem k_2(n) (cf. A001197 and other crossrefs) which asks the converse: how many 1's must be in an n X n {0,1}-matrix in order to guarantee the existence of an all-ones 2 X 2 submatrix. This complementarity leads to the given formula which was used to compute the given values.

See A347473 and A347474 for the similar problem with a 3 X 3 resp. 4 X 4 zero submatrix.

Examples

			For n = 2, there must not be any nonzero entry in an n X n = 2 X 2 matrix, if one wants a 2 X 2 zero submatrix, whence a(2) = 0.
For n = 3, having at most 2 nonzero entries in the n X n matrix still guarantees that there is a 2 X 2 zero submatrix (delete the row of the first nonzero entry and then the column of the remaining nonzero entry, if any), but if one allows 3 nonzero entries and they are placed on the diagonal, then there is no 2 X 2 zero submatrix. Hence, a(3) = 2.
		

Crossrefs

Cf. A001197 (k_2(n)), A001198 (k_3(n)), A006613 - A006626.

Cf. A347473, A347474 (analog for 3 X 3 resp. 4 X 4 zero submatrix).

Cf. A350296.

Formula

a(n) = n^2 - A001197(n).

a(n) = A350296(n) - 1. - Andrew Howroyd, Dec 23 2021

Extensions

a(22)-a(24) computed from A001197 by Max Alekseyev, Feb 08 2022