

A287383


Triangle read by rows: T(n,m) is the number of inequivalent n X m matrices under action of the Klein group, with oneninth 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).


8



1, 1, 1, 1, 1, 1, 1, 1, 1, 90720, 1, 1, 1, 14968800, 40864824000, 1, 1, 453600, 5108114880, 131993382447360, 3463115239584000000
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OFFSET

0,10


COMMENTS

Computed using Polya's enumeration theorem for coloring.


LINKS



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*nm)/2) + 2*y2^(m*n/2))/4 for odd n and even m; (y1^(m*n) + y1^m*y2^((m*nn)/2) + 2*y2^(m*n/2))/4 for even n and odd m; (y1^(m*n) + y1^n*y2^((m*nn)/2) + y1^m*y2^((m*nm)/2) + y1*y2^((m*n1)/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 (9k) numbers, if m*n = k mod 9.


EXAMPLE

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


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KEYWORD



AUTHOR



STATUS

approved



