A287020 Triangle read by rows: T(n,m) is the number of inequivalent n X m matrices under action of the Klein group, with one-fourth each of 1s, 2s, 3s and 4s (ordered occurrences rounded up/down if n*m != 0 mod 4).

1, 1, 1, 1, 1, 6, 1, 1, 46, 1926, 1, 12, 648, 92544, 15767640, 1, 30, 6312, 3943710, 2933201376, 2061379857600, 1, 90, 92400, 192994200, 577186150464, 1605824110657800, 5363188066566330000, 1, 318, 1051140, 10266445476, 118129589107200, 1340797019145183600

Offset: 0

Author

María Merino, Imanol Unanue, May 18 2017

Keywords

• nonn
• tabl

Comments

Computed using Polya's enumeration theorem for coloring.

Examples

			For n = 4 and m = 2 the T(4,2) = 648 solutions are colorings of 4 X 2 matrices in 4 colors inequivalent under the action of the Klein group with exactly 2 occurrences of each color (coefficient of x1^2 x2^2 x3^2 x4^2).
Triangle begins:
========================================================
n\m | 0   1    2     3         4            5
----|---------------------------------------------------
0   | 1
1   | 1   1
2   | 1   1    6
3   | 1   1    46    1926
4   | 1   12   648   92544     15767640
5   | 1   30   6312  3943710   2933201376   2061379857600
		

Links

• María Merino, Rows n=0..42 of triangle, flattened
• M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Crossrefs

Cf. A283435, A286892, A287021, A287022, A287377, A287378, A287383, A287384.

Formula

G.f.: g(x1,x2,x3,x4)=(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+x4, y2=x1^2+x2^2+x3^2+x4^2, and occurrences of numbers are ceiling(m*n/4) for the first k numbers and floor(m*n/4) for the last (4-k) numbers, if m*n = k mod 4.