A286892 Triangle read by rows: T(n,m) is the number of inequivalent n X m matrices under action of the Klein group, with one-third each of 1s, 2s and 3s (ordered occurrences rounded up/down if m*n != 0 mod 3).
1, 1, 1, 1, 1, 3, 1, 3, 27, 438, 1, 6, 140, 8766, 504504, 1, 16, 1056, 189774, 33258880, 6573403050, 1, 48, 8730, 4292514, 2366403930, 1387750992012, 846182953495152, 1, 108, 63108, 99797220, 159511561440, 282061024690536, 530143167401850960, 976645996512669379710
Offset: 0
Examples
For n = 3 and m = 2 the T(3,2) = 27 solutions are colorings of 3 X 2 matrices in 3 colors inequivalent under the action of the Klein group with exactly 2 occurrences of each color (coefficient of x1^2 x2^2 x3^2). Triangle begins: ================================================= n\m | 0 1 2 3 4 5 ----|-------------------------------------------- 0 | 1 1 | 1 1 2 | 1 1 3 3 | 1 3 27 438 4 | 1 6 140 8766 504504 5 | 1 16 1056 189774 33258880 6573403050
Links
- María Merino, Rows n=0..47 of triangle, flattened
- M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).
Formula
G.f.: g(x1,x2,x3)=(y1^(m*n) + 3*y2^(m*n/2))/4 for even n and m;
(y1^(m*n) + y1^n*y2^((m*n-m)/2) + 2*y2^(m*n/2))/4 for odd n and even m;
(y1^(m*n) + y1^m*y2^((m*n-n)/2) + 2*y2^(m*n/2))/4 for even n and odd m;
(y1^(m*n) + y1^n*y2^((m*n-n)/2) + y1^m*y2^((m*n-m)/2) + y1*y2^((m*n-1)/2))/4 for odd n and m; where coefficient correspond to y1=x1+x2+x3, y2=x1^2+x2^2+x3^2, and occurrences of numbers are ceiling(m*n/3) for the first k numbers and floor(m*n/3) for the last (3-k) numbers, if m*n = k mod 3.
Comments