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A287020 Triangle read by rows: T(n,m) is the number of inequivalent n X m matrices under action of the Klein group, with one-fourth each of 1s, 2s, 3s and 4s (ordered occurrences rounded up/down if n*m != 0 mod 4). 9
1, 1, 1, 1, 1, 6, 1, 1, 46, 1926, 1, 12, 648, 92544, 15767640, 1, 30, 6312, 3943710, 2933201376, 2061379857600, 1, 90, 92400, 192994200, 577186150464, 1605824110657800, 5363188066566330000, 1, 318, 1051140, 10266445476, 118129589107200, 1340797019145183600 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Computed using Polya's enumeration theorem for coloring.

LINKS

María Merino, Rows n=0..42 of triangle, flattened

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

FORMULA

G.f.: g(x1,x2,x3,x4)=(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=x1+x2+x3+x4, y2=x1^2+x2^2+x3^2+x4^2, and occurrences of numbers are ceiling(m*n/4) for the first k numbers and floor(m*n/4) for the last (4-k) numbers, if m*n = k mod 4.

EXAMPLE

For n = 4 and m = 2 the T(4,2) = 648 solutions are colorings of 4 X 2 matrices in 4 colors inequivalent under the action of the Klein group with exactly 2 occurrences of each color (coefficient of x1^2 x2^2 x3^2 x4^2).

Triangle begins:

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

n\m | 0 1 2 3 4 5

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

0 | 1

1 | 1 1

2 | 1 1 6

3 | 1 1 46 1926

4 | 1 12 648 92544 15767640

5 | 1 30 6312 3943710 2933201376 2061379857600

CROSSREFS

Cf. A283435, A286892, A287021, A287022, A287377, A287378, A287383, A287384.

Sequence in context: A156764 A156765 A015117 * A172375 A075377 A046792

Adjacent sequences: A287017 A287018 A287019 * A287021 A287022 A287023

KEYWORD

nonn,tabl

AUTHOR

María Merino, Imanol Unanue, May 18 2017

STATUS

approved

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Last modified January 29 00:02 EST 2023. Contains 359905 sequences. (Running on oeis4.)