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
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
Formula
Extensions
a(8) from Brendan McKay, May 25 2020
Comments