A283435 Triangle read by rows: T(n,m) is the number of binary pattern classes in the (n,m)-rectangular grid with half 1's and half 0's: two patterns are in same class if one can be obtained by a reflection or 180-degree rotation of the other (ordered occurrences rounded up/down if m*n is odd).
1, 1, 1, 1, 1, 3, 1, 2, 6, 39, 1, 4, 22, 252, 3270, 1, 6, 66, 1675, 46448, 1302196, 1, 10, 246, 12300, 676732, 38786376, 2268820290, 1, 19, 868, 88900, 10032648, 1134474924, 134564842984, 15801337532526
Offset: 0
Examples
For n = 3 and m = 2 the T(3,2) = 6 solutions are colorings of 3 X 2 matrices in 2 colors inequivalent under the action of the Klein group with exactly 3 occurrences of each color (coefficient of x1^3 x2^3). Triangle begins: ====================================== n\m | 0 1 2 3 4 5 ----|--------------------------------- 0 | 1 1 | 1 1 2 | 1 1 3 3 | 1 2 6 39 4 | 1 4 22 252 3270 5 | 1 6 66 1675 46448 1302196
Links
- María Merino, Rows n=0..59 of triangle, flattened
- María Merino and Imanol Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).
Formula
G.f.: g(x1,x2)=(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, y2=x1^2+x2^2 and occurrences of numbers are ceiling(m*n/2) for 0's and floor(m*n/2) for 1's.
Comments