A287384 Triangle read by rows: T(n,m) is the number of inequivalent n X m matrices under action of the Klein group, with one-tenth each of 1's, 2's, 3's, 4's, 5's, 6's, 7's, 8's, 9's and 0's (ordered occurrences rounded up/down if n*m != 0 mod 10).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 29937600, 81729648000

Offset

0,14

Comments

Computed using Polya s enumeration theorem for coloring.

Links

María Merino, Rows n=0..32 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,x5,x6,x7,x8,x9,x10)=(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..10} x_i, y2=Sum_{i=1..10} x_i^2, and occurrences of numbers are ceiling(m*n/10) for the first k numbers and floor(m*n/10) for the last (10-k) numbers, if m*n = k mod 10.

Example

For n = 4 and m = 3 the T(4,3)=29937600 solutions are colorings of 4 X 3 matrices in 10 colors inequivalent under the action of the Klein group with exactly 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 occurrences of each color (coefficient of x1^2 x2^2 x3^1 x4^1 x5^1 x6^1 x7^1 x8^1 x9^1 x10^1).
Triangle begins:
==========================================
n\m | 0  1  2   3           4
----|-------------------------------------
0   | 1
1   | 1  1
2   | 1  1  1
3   | 1  1  1   1
4   | 1  1  1   29937600    81729648000

Crossrefs

Cf. A283435, A286892, A287020, A287021, A287022, A287377, A287378, A287383.

Sequence in context: A176141 A105013 A168515 * A107619 A320723 A187431

Adjacent sequences: A287381 A287382 A287383 * A287385 A287386 A287387

Keyword

nonn,tabl

Author

María Merino, Imanol Unanue, May 24 2017

Status

approved