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A116976 Number of nonsingular n X n matrices with rational entries equal to 0 or 1, up to row and column permutations.

%I A116976 #28 Jul 14 2022 13:58:11
%S A116976 1,2,8,61,1153,64310,11352457,6417769762
%N A116976 Number of nonsingular n X n matrices with rational entries equal to 0 or 1, up to row and column permutations.
%C A116976 "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 A116976 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 A116976 <a href="/index/Mat#binmat">Index entries for sequences related to binary matrices</a>
%F A116976 a(n) = A002724(n) - A116977(n). - _Max Alekseyev_, Jul 14 2022
%e A116976 From _M. F. Hasler_, May 25 2020: (Start)
%e A116976 Representatives of the two inequivalent nonsingular (0,1) matrices for n=2 are
%e A116976   [ 1  0 ]   and   [ 1  1 ]  .
%e A116976   [ 0  1 ]         [ 0  1 ]
%e A116976 For n=3 we have 8 nonsingular nonequivalent representatives:
%e A116976   [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 A116976   [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 A116976   [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 A116976 To see that they are inequivalent, consider their column sums:
%e A116976   (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 A116976 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 A116976 (End)
%Y A116976 Cf. A055165, A002724.
%K A116976 nonn,hard,more
%O A116976 1,2
%A A116976 _Vladeta Jovovic_, Apr 01 2006
%E A116976 a(8) from _Brendan McKay_, May 25 2020