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 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: A296138 A350542 A159951 * A013862 A116233 A350543 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

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Last modified May 31 21:03 EDT 2023. Contains 363068 sequences. (Running on oeis4.)