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%I #43 Jan 03 2021 21:39:18
%S 1,12,1330560
%N The number of (n^2) X (n^2) real {0,1}-matrices the square of which is the all-ones matrix.
%C These are real {0,1} matrices A such that A^2 = J, the all-ones matrix.
%C 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.
%C 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.
%C a(3) = 1330560 is confirmed by _W. Edwin Clark_, Mar 12 2017, who says: (Start)
%C 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.
%C 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.
%C (End)
%H F. Curtis, J. Drew, <a href="http://www.resnet.wm.edu/~cklixx/reu02.pdf">Central groupoids, central digraphs, and zero-one matrices A satisfying A^2=J</a>, (2002).
%H F. Curtis, J. Drew, C-K Li, D. Pragel, <a href="http://dx.doi.org/10.1016/j.jcta.2003.10.001">Central groupoids, central digraphs, and zero-one matrices A satisfying A^2=J</a>, J. Combin. Theo. A (105) (2004) 35-50.
%H J. Knuth, <a href="http://dx.doi.org/10.1016/S0021-9800(70)80032-1">Notes on Central Groupoids</a>, J. Combin. Theo. 8 (1970) 376-390.
%H Donald E. Knuth and Peter B. Bendix, <a href="http://www.cs.tufts.edu/~nr/cs257/archive/don-knuth/knuth-bendix.pdf">Simple word problems in universal algebras</a>, in John Leech (ed.), Computational Problems in Abstract Algebra, Pergamon, 1970, pp. 263-297.
%H H. Ryser, <a href="http://dx.doi.org/10.1016/0024-3795(70)90036-4">A generalization of the matrix equation A^2=J</a>, Lin. Algebra Applic. 3 (4) (1970) 451-460.
%H Y.-K. Wu, R.-Z. Jia, Q. Li, <a href="http://dx.doi.org/10.1016/S0024-3795(01)00491-8">g-circulant solutions to the (0,1) matrix equation A^m=J_n</a>, Lin. Alg. Applic. 345 (1-3) (2002) 195-224.
%e Four of the 12 solutions for 4 X 4 are
%e 1 1 0 0
%e 0 0 1 1
%e 1 1 0 0
%e 0 0 1 1
%e .
%e 1 1 0 0
%e 0 0 1 1
%e 0 0 1 1
%e 1 1 0 0
%e .
%e 1 0 0 1
%e 0 1 1 0
%e 1 0 0 1
%e 0 1 1 0
%e .
%e 0 1 0 1
%e 1 0 1 0
%e 1 0 1 0
%e 0 1 0 1
%e .
%e Solutions for 9 X 9 are, for example,
%e 1 1 1 0 0 0 0 0 0
%e 0 0 0 1 0 1 1 0 0
%e 0 0 0 0 1 0 0 1 1
%e 1 1 1 0 0 0 0 0 0
%e 1 1 1 0 0 0 0 0 0
%e 0 0 0 1 0 1 1 0 0
%e 0 0 0 0 1 0 0 1 1
%e 0 0 0 1 0 1 1 0 0
%e 0 0 0 0 1 0 0 1 1
%e .
%e 1 1 1 0 0 0 0 0 0
%e 0 0 0 1 0 1 1 0 0
%e 0 0 0 0 1 0 0 1 1
%e 1 1 1 0 0 0 0 0 0
%e 1 1 1 0 0 0 0 0 0
%e 0 0 0 1 0 1 1 0 0
%e 0 0 0 0 1 0 0 1 1
%e 0 0 0 1 0 0 0 1 1
%e 0 0 0 0 1 1 1 0 0
%e .
%e 1 1 1 0 0 0 0 0 0
%e 0 0 0 1 0 1 1 0 0
%e 0 0 0 0 1 0 0 1 1
%e 1 1 1 0 0 0 0 0 0
%e 1 1 1 0 0 0 0 0 0
%e 0 0 0 1 0 1 0 0 1
%e 0 0 0 0 1 0 1 1 0
%e 0 0 0 1 0 1 0 0 1
%e 0 0 0 0 1 0 1 1 0
%Y Cf. A008300.
%Y See also A283643.
%K nonn,bref,more
%O 1,2
%A _R. J. Mathar_, Mar 12 2017
%E Edited by _N. J. A. Sloane_, Mar 12 2017