login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A283627 The number of (n^2) X (n^2) real {0,1}-matrices the square of which is the all-ones matrix. 2
1, 12, 1330560 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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)

LINKS

Table of n, a(n) for n=1..3.

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.

EXAMPLE

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

CROSSREFS

Cf. A008300.

See also A283643.

Sequence in context: A055312 A296138 A159951 * A013862 A116233 A145745

Adjacent sequences:  A283624 A283625 A283626 * A283628 A283629 A283630

KEYWORD

nonn,bref,more

AUTHOR

R. J. Mathar, Mar 12 2017

EXTENSIONS

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

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 27 11:33 EDT 2021. Contains 348276 sequences. (Running on oeis4.)