A287377 - OEIS

A287377

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).

1, 1, 1, 1, 1, 1, 1, 1, 1, 22680, 1, 1, 5040, 3742560, 4540536000, 1, 1, 113400, 851370480, 6518191680000, 54111175679736000

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OFFSET

0,10

COMMENTS

Computed using Polya's enumeration theorem for coloring.

LINKS

María Merino, Rows n=0..36 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)=(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 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.

EXAMPLE

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).

Triangle begins:

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

n\m | 0 1 2 3 4 5

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

0 | 1

1 | 1 1

2 | 1 1 1

3 | 1 1 1 22680

4 | 1 1 5040 3742560 4540536000

5 | 1 1 113400 851370480 6518191680000 54111175679736000

CROSSREFS

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