A287378 - OEIS

%I #29 Apr 29 2019 05:19:24

%S 1,1,1,1,1,1,1,1,1,45360,1,1,10080,7484544,20432442240,1,1,226800,

%T 2554075440,29331862801920,577185873264000000

%N 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).

%C Computed using Polya's enumeration theorem for coloring.

%H María Merino, <a href="/A287378/b287378.txt">Rows n=0..35 of triangle, flattened</a>

%H M. Merino and I. Unanue, <a href="https://doi.org/10.1387/ekaia.17851">Counting squared grid patterns with Pólya Theory</a>, EKAIA, 34 (2018), 289-316 (in Basque).

%F g(x1,x2,x3,x4,x5,x6,x7,x8) = (y1^(m*n) + 3*y2^(m*n/2))/4 for even n and m;

%F (y1^(m*n) + y1^n*y2^((m*n-m)/2) + 2*y2^(m*n/2))/4 for odd n and even m;

%F (y1^(m*n) + y1^m*y2^((m*n-n)/2) + 2*y2^(m*n/2))/4 for even n and odd m;

%F (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.

%e 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).

%e Triangle begins:

%e =================================================================

%e n\m | 0  1  2       3           4               5

%e ----|------------------------------------------------------------

%e 0   | 1

%e 1   | 1  1

%e 2   | 1  1  1

%e 3   | 1  1  1       45360

%e 4   | 1  1  10080   7484544     20432442240

%e 5   | 1  1  226800  2554075440  29331862801920  577185873264000000

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

%K nonn,tabl

%O 0,10

%A _María Merino_, Imanol Unanue, May 24 2017