A309244 Triangle of number of nonsingular n X n matrices over GF(2) by number of ones.

1, 0, 2, 4, 0, 0, 6, 36, 72, 36, 18, 0, 0, 0, 24, 288, 1440, 3648, 4752, 4992, 2592, 1728, 600, 96, 0, 0, 0, 0, 120, 2400, 21600, 112800, 369600, 808800, 1384800, 1663200, 1849200, 1466400, 1143840, 636000, 345600, 141600, 45600, 7200, 600, 0, 0, 0, 0, 0, 720

Offset: 1

Author

Brian Hopkins, Jul 17 2019

Keywords

• nonn
• tabf

Comments

The row for n begins with n-1 zeros since a matrix with fewer than n ones has an all-zero row.

The last entry in the row for n is T(n, n^2-n+1) as a matrix with more than n^2-n+1 ones must have two identical rows.

Each entry in the row for n is a multiple of n! since rows must be distinct.

Examples

			T(2,3) = 4 from the 2 X 2 nonsingular matrices (1,1;1,0), (1,1;0,1), (1,0;1,1), and (0,1;1,1) which each have 3 ones.
Triangle begins
1
0 2 4
0 0 6 36  72   36   18
0 0 0 24 288 1440 3648 4752 4992 2592 1728 600 96
		

Links

• Mathoverflow, The number of non-singular n x n matrices over F2 with exactly k non-zero entries, posted 12 Jun 2019. Rows for n = 3 and n = 4 given by Richard Stanley in a comment.

Crossrefs

Row sums are A002884.

Formula

T(n, n) = n!, T(n, n+1) = n!*n*(n-1), T(n, n^2-n+1) = n!*n (Weg, see Mathoverflow link).