A287020 - OEIS

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

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OFFSET

0,6

COMMENTS

Computed using Polya's enumeration theorem for coloring.

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).

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.

EXAMPLE

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

CROSSREFS

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