A309244 - OEIS

%I #21 Jul 19 2019 11:12:15

%S 1,0,2,4,0,0,6,36,72,36,18,0,0,0,24,288,1440,3648,4752,4992,2592,1728,

%T 600,96,0,0,0,0,120,2400,21600,112800,369600,808800,1384800,1663200,

%U 1849200,1466400,1143840,636000,345600,141600,45600,7200,600,0,0,0,0,0,720

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

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

%C 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.

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

%H Mathoverflow, <a href="https://mathoverflow.net/questions/333837/the-number-of-non-singular-n-times-n-matrices-over-mathbbf-2-with-exactly">The number of non-singular n x n matrices over F2 with exactly k non-zero entries</a>, posted 12 Jun 2019.  Rows for n = 3 and n = 4 given by Richard Stanley in a comment.

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

%e 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.

%e Triangle begins

%e 1

%e 0 2 4

%e 0 0 6 36  72   36   18

%e 0 0 0 24 288 1440 3648 4752 4992 2592 1728 600 96

%Y Row sums are A002884.

%K nonn,tabf

%O 1,3

%A _Brian Hopkins_, Jul 17 2019