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A286892
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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).
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9
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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
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OFFSET
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0,6
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COMMENTS
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Computed using Polya's enumeration theorem for coloring.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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