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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.
2
1, 2, 8, 61, 1153, 64310, 11352457, 6417769762
OFFSET
1,2
COMMENTS
"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
LINKS
Miodrag Zivkovic, Classification of small (0,1) matrices, Linear Algebra and its Applications, 414 (2006), 310-346. See also on arXiv, arXiv:math/0511636 [math.CO], 2005.
FORMULA
a(n) = A002724(n) - A116977(n). - Max Alekseyev, Jul 14 2022
EXAMPLE
From M. F. Hasler, May 25 2020: (Start)
Representatives of the two inequivalent nonsingular (0,1) matrices for n=2 are
[ 1 0 ] and [ 1 1 ] .
[ 0 1 ] [ 0 1 ]
For n=3 we have 8 nonsingular nonequivalent representatives:
[1 0 0] [1 0 0] [1 0 1] [1 1 1] [1 1 0] [1 1 0] [1 1 1] [1 1 0]
[0 1 0], [0 1 1], [0 1 1], [0 1 0], [0 1 1], [1 0 1], [0 1 1], [1 0 1].
[0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 1 1] [0 0 1] [1 1 1]
To see that they are inequivalent, consider their column sums:
(1 1 1), (1 1 2), (1 1 3), (1 2 2), (1 2 2), (2 2 2), (1 2 3), (3 2 2).
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.
(End)
CROSSREFS
Sequence in context: A140722 A327078 A332779 * A132574 A086903 A161566
KEYWORD
nonn,hard,more
AUTHOR
Vladeta Jovovic, Apr 01 2006
EXTENSIONS
a(8) from Brendan McKay, May 25 2020
STATUS
approved