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

1,2

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

Table of n, a(n) for n=1..8.

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

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

Cf. A055165, A002724.

Sequence in context: A140722 A327078 A332779 * A132574 A086903 A161566

Adjacent sequences: A116973 A116974 A116975 * A116977 A116978 A116979

Keyword

nonn,hard,more

Author

Vladeta Jovovic, Apr 01 2006

Extensions

a(8) from Brendan McKay, May 25 2020

Status

approved