A287021 Triangle read by rows: T(n,m) is the number of inequivalent n X m matrices under action of the Klein group, with one-fifth of 1s, 2s, 3s, 4s and 5s (ordered occurrences rounded up/down if n*m != 0 mod 5).
1, 1, 1, 1, 1, 1, 1, 1, 90, 5712, 1, 1, 1260, 416064, 168168000, 1, 60, 28440, 42045600, 76385194200, 155840192585280, 1, 180, 415800, 3216282300, 31168037156256, 342718542439257600, 3574641463338838464000, 1, 630, 8408400, 320818773240, 14181456923282880, 794364769671213312000, 40694019408428534970822000, 2416738787895064029335795945088
Offset: 0
Examples
For n = 5 and m = 2 the T(5,2) = 28440 solutions are colorings of 5 X 2 matrices in 5 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 x5^2). Triangle begins: ============================================================ n\m | 0 1 2 3 4 5 ----|------------------------------------------------------- 0 | 1 1 | 1 1 2 | 1 1 1 3 | 1 1 90 5712 4 | 1 1 1260 416064 168168000 5 | 1 60 28440 42045600 76385194200 155840192585280
Links
- María Merino, Rows n=0..38 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)=(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..5} x_i, y2=Sum_{i=1..5} x_i^2, and occurrences of numbers are ceiling(m*n/5) for the first k numbers and floor(m*n/5) for the last (5-k) numbers, if m*n = k mod 5.
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