|
| |
|
|
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).
|
|
8
|
|
|
|
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 29937600, 81729648000
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,14
|
|
|
COMMENTS
|
Computed using Polya s enumeration theorem for coloring.
|
|
|
LINKS
|
|
|
|
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
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
