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

1, 2, 8, 61, 1153, 64310, 11352457, 6417769762

Offset: 1

Author

Vladeta Jovovic, Apr 01 2006

Keywords

• nonn
• hard
• more

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

Examples

			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)
		

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.
• Index entries for sequences related to binary matrices

Crossrefs

Cf. A055165, A002724.

Formula

a(n) = A002724(n) - A116977(n). - Max Alekseyev, Jul 14 2022

Extensions

a(8) from Brendan McKay, May 25 2020