A287383 Triangle read by rows: T(n,m) is the number of inequivalent n X m matrices under action of the Klein group, with one-ninth each of 1's, 2's, 3's, 4's, 5's, 6's, 7's, 8's and 9's (ordered occurrences rounded up/down if n*m != 0 mod 9).
1, 1, 1, 1, 1, 1, 1, 1, 1, 90720, 1, 1, 1, 14968800, 40864824000, 1, 1, 453600, 5108114880, 131993382447360, 3463115239584000000
Offset: 0
Examples
For n = 3 and m = 3 the T(3,3) = 90720 solutions are colorings of 3 X 3 matrices in 9 colors inequivalent under the action of the Klein group with exactly 1 occurrence of each color (coefficient of x1^1, x2^1, x3^1, x4^1, x5^1, x6^1, x7^1, x8^1, x9^1). Triangle begins: =================================================================== n\m | 0 1 2 3 4 5 ----|-------------------------------------------------------------- 0 | 1 1 | 1 1 2 | 1 1 1 3 | 1 1 1 90720 4 | 1 1 1 14968800 40864824000 5 | 1 1 453600 5108114880 131993382447360 3463115239584000000
Links
- María Merino, Rows n=0..34 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,x5,x6,x7,x8,x9) = (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 the coefficients y1 and y2 correspond to y1 = Sum_{i=1..9} x_i and y2 = Sum_{i=1..9} x_i^2. Occurrences of numbers are ceiling(m*n/9) for the first k numbers and floor(m*n/9) for the last (9-k) numbers, if m*n = k mod 9.
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