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%I #28 Jul 14 2022 13:58:11
%S 1,2,8,61,1153,64310,11352457,6417769762
%N Number of nonsingular n X n matrices with rational entries equal to 0 or 1, up to row and column permutations.
%C "Rational entries" means that a matrix is nonsingular iff it has a nonzero determinant. (Over the integers a matrix with determinant > 1 is not invertible.) _M. F. Hasler_, May 25 2020
%H Miodrag Zivkovic, <a href="https://doi.org/10.1016/j.laa.2005.10.010">Classification of small (0,1) matrices</a>, Linear Algebra and its Applications, 414 (2006), 310-346. See also on <a href="https://arxiv.org/abs/math/0511636">arXiv</a>, arXiv:math/0511636 [math.CO], 2005.
%H <a href="/index/Mat#binmat">Index entries for sequences related to binary matrices</a>
%F a(n) = A002724(n) - A116977(n). - _Max Alekseyev_, Jul 14 2022
%e From _M. F. Hasler_, May 25 2020: (Start)
%e Representatives of the two inequivalent nonsingular (0,1) matrices for n=2 are
%e [ 1 0 ] and [ 1 1 ] .
%e [ 0 1 ] [ 0 1 ]
%e For n=3 we have 8 nonsingular nonequivalent representatives:
%e [1 0 0] [1 0 0] [1 0 1] [1 1 1] [1 1 0] [1 1 0] [1 1 1] [1 1 0]
%e [0 1 0], [0 1 1], [0 1 1], [0 1 0], [0 1 1], [1 0 1], [0 1 1], [1 0 1].
%e [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 1 1] [0 0 1] [1 1 1]
%e To see that they are inequivalent, consider their column sums:
%e (1 1 1), (1 1 2), (1 1 3), (1 2 2), (1 2 2), (2 2 2), (1 2 3), (3 2 2).
%e Only the 4th and 5th matrix have equivalent column sum signature (1,2,2), but their row sums are (3,1,1) resp. (2,2,1). Therefore they can't be obtained one from the other by row and column permutations which leave invariant these sums.
%e (End)
%Y Cf. A055165, A002724.
%K nonn,hard,more
%O 1,2
%A _Vladeta Jovovic_, Apr 01 2006
%E a(8) from _Brendan McKay_, May 25 2020