A283627 The number of (n^2) X (n^2) real {0,1}-matrices the square of which is the all-ones matrix.

1, 12, 1330560

Offset: 1

Author

R. J. Mathar, Mar 12 2017

Keywords

• nonn
• bref
• more

Comments

These are real {0,1} matrices A such that A^2 = J, the all-ones matrix.

It is known that A must have dimension which is a square, so the sequence shows counts for 1 X 1, 4 X 4, 9 X 9, 16 X 16 matrices and so on.

It is also known that if A has dimension n^2 X n^2, then each row and column must contain exactly n 1's, and A has trace n.

a(3) = 1330560 is confirmed by W. Edwin Clark, Mar 12 2017, who says: (Start)

My method was to take the 6 matrices A1, A2, A3, A4, A5, A6 found by Knuth, which are representatives for the 6 distinct orbits of 9 x 9 matrices A such that A^2 = J under the action of the 9! permutation matrices acting by conjugation.

I found for each Ai the size n_i of the stabilizer of Ai. The stabilizer orders are [n_1,n_2,n_3,n_4,n_5,n_6] = [6,2,1,1,2,2], which implies that the cardinality of the union of all orbits is Sum(9!/n_i, i=1..6) = 1330560.

(End)

Examples

			Four of the 12 solutions for 4 X 4 are
  1 1 0 0
  0 0 1 1
  1 1 0 0
  0 0 1 1
.
  1 1 0 0
  0 0 1 1
  0 0 1 1
  1 1 0 0
.
  1 0 0 1
  0 1 1 0
  1 0 0 1
  0 1 1 0
.
  0 1 0 1
  1 0 1 0
  1 0 1 0
  0 1 0 1
.
Solutions for 9 X 9 are, for example,
  1 1 1 0 0 0 0 0 0
  0 0 0 1 0 1 1 0 0
  0 0 0 0 1 0 0 1 1
  1 1 1 0 0 0 0 0 0
  1 1 1 0 0 0 0 0 0
  0 0 0 1 0 1 1 0 0
  0 0 0 0 1 0 0 1 1
  0 0 0 1 0 1 1 0 0
  0 0 0 0 1 0 0 1 1
.
  1 1 1 0 0 0 0 0 0
  0 0 0 1 0 1 1 0 0
  0 0 0 0 1 0 0 1 1
  1 1 1 0 0 0 0 0 0
  1 1 1 0 0 0 0 0 0
  0 0 0 1 0 1 1 0 0
  0 0 0 0 1 0 0 1 1
  0 0 0 1 0 0 0 1 1
  0 0 0 0 1 1 1 0 0
.
  1 1 1 0 0 0 0 0 0
  0 0 0 1 0 1 1 0 0
  0 0 0 0 1 0 0 1 1
  1 1 1 0 0 0 0 0 0
  1 1 1 0 0 0 0 0 0
  0 0 0 1 0 1 0 0 1
  0 0 0 0 1 0 1 1 0
  0 0 0 1 0 1 0 0 1
  0 0 0 0 1 0 1 1 0
		

Links

• F. Curtis, J. Drew, Central groupoids, central digraphs, and zero-one matrices A satisfying A^2=J, (2002).
• F. Curtis, J. Drew, C-K Li, D. Pragel, Central groupoids, central digraphs, and zero-one matrices A satisfying A^2=J, J. Combin. Theo. A (105) (2004) 35-50.
• J. Knuth, Notes on Central Groupoids, J. Combin. Theo. 8 (1970) 376-390.
• Donald E. Knuth and Peter B. Bendix, Simple word problems in universal algebras, in John Leech (ed.), Computational Problems in Abstract Algebra, Pergamon, 1970, pp. 263-297.
• H. Ryser, A generalization of the matrix equation A^2=J, Lin. Algebra Applic. 3 (4) (1970) 451-460.
• Y.-K. Wu, R.-Z. Jia, Q. Li, g-circulant solutions to the (0,1) matrix equation A^m=J_n, Lin. Alg. Applic. 345 (1-3) (2002) 195-224.

Crossrefs

Cf. A008300.

See also A283643.

Extensions

Edited by N. J. A. Sloane, Mar 12 2017