|
| |
|
|
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).
|
|
9
|
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,9
|
|
|
COMMENTS
|
Computed using Polya's enumeration theorem for coloring.
|
|
|
LINKS
|
|
|
|
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.
|
|
|
EXAMPLE
|
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
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
