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

1,3

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.

Links

Table of n, a(n) for n=1..51.

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.

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).

Example

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

Crossrefs

Row sums are A002884.

Sequence in context: A072069 A230423 A213672 * A004025 A102561 A072068

Adjacent sequences: A309241 A309242 A309243 * A309245 A309246 A309247

Keyword

nonn,tabf

Author

Brian Hopkins, Jul 17 2019

Status

approved