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, 1, 1, 907200, 10216209600, 593970221376000, 15584018578345728000, 1, 1, 29937600, 6252318979200, 1870082229375360000, 1096699334071461120000000, 375493744214599112902800000000

Offset: 0

Author

María Merino and Imanol Unanue, May 24 2017

Keywords

• nonn
• tabl

Comments

Computed using Polya's enumeration theorem for coloring.

Examples

			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
		

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).

Crossrefs

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

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.

Extensions

More terms from María Merino, Aug 01 2017