A287377 - OEIS

%I #24 Apr 10 2020 01:39:42

%S 1,1,1,1,1,1,1,1,1,22680,1,1,5040,3742560,4540536000,1,1,113400,

%T 851370480,6518191680000,54111175679736000

%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-seventh each of 1's, 2's, 3's, 4's, 5's, 6's and 7's (ordered occurrences rounded up/down if n*m != 0 mod 7).

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

%H María Merino, <a href="/A287377/b287377.txt">Rows n=0..36 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)=(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; (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=Sum_{i=1..7} x_i, y2=Sum_{i=1..7} x_i^2, and occurrences of numbers are ceiling(m*n/7) for the first k numbers and floor(m*n/7) for the last (7-k) numbers, if m*n = k mod 7.

%e For n = 4 and m = 2 the T(4,2) = 5040 solutions are colorings of 4 X 2 matrices in 7 colors inequivalent under the action of the Klein group with exactly 2, 1, 1, 1, 1, 1, 1 occurrences of each color (coefficient of x1^2 x2^1 x3^1 x4^1 x5^1 x6^1 x7^1).

%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       22680

%e 4   | 1  1  5040    3742560    4540536000

%e 5   | 1  1  113400  851370480  6518191680000  54111175679736000

%Y Cf. A283435, A286892, A287020, A287021, A287022, A287378, A287383, A287384.

%K nonn,tabl

%O 0,10

%A _María Merino_, Imanol Unanue, May 24 2017