

A283627


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


2




OFFSET

1,2


COMMENTS

These are real {0,1} matrices A such that A^2 = J, the allones 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 zeroone matrices A satisfying A^2=J, (2002).
F. Curtis, J. Drew, CK Li, D. Pragel, Central groupoids, central digraphs, and zeroone matrices A satisfying A^2=J, J. Combin. Theo. A (105) (2004) 3550.
J. Knuth, Notes on Central Groupoids, J. Combin. Theo. 8 (1970) 376390.
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. 263297.
H. Ryser, A generalization of the matrix equation A^2=J, Lin. Algebra Applic. 3 (4) (1970) 451460.
Y.K. Wu, R.Z. Jia, Q. Li, gcirculant solutions to the (0,1) matrix equation A^m=J_n, Lin. Alg. Applic. 345 (13) (2002) 195224.


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



