A287378 - OEIS

A287378

Triangle read by rows: T(n,m) is the number of inequivalent n X m matrices under action of the Klein group, with one-eighth each of 1's, 2's, 3's, 4's, 5's, 6's, 7's and 8's (ordered occurrences rounded up/down if n*m != 0 mod 8).

1, 1, 1, 1, 1, 1, 1, 1, 1, 45360, 1, 1, 10080, 7484544, 20432442240, 1, 1, 226800, 2554075440, 29331862801920, 577185873264000000

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OFFSET

0,10

COMMENTS

Computed using Polya's enumeration theorem for coloring.

LINKS

María Merino, Rows n=0..35 of triangle, flattened

M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

FORMULA

g(x1,x2,x3,x4,x5,x6,x7,x8) = (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..8} x_i and y2 = Sum_{i=1..8} x_i^2. Occurrences of numbers are ceiling(m*n/8) for the first k numbers and floor(m*n/8) for the last (8-k) numbers, if m*n = k mod 8.

EXAMPLE

For n = 4 and m = 2, the T(4,2) = 10080 solutions are colorings of 4 X 2 matrices in 8 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).

Triangle begins:

=================================================================

n\m | 0 1 2 3 4 5

----|------------------------------------------------------------

0 | 1

1 | 1 1

2 | 1 1 1

3 | 1 1 1 45360

4 | 1 1 10080 7484544 20432442240

5 | 1 1 226800 2554075440 29331862801920 577185873264000000

CROSSREFS

Cf. A283435, A286892, A287020, A287021, A287022, A287377, A287383, A287384.