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A225542
Number T(n,k,u) of partitions of an n X k rectangle into integer-sided square parts containing u nodes that are unconnected to any of their neighbors, considering only the number of parts; irregular triangle T(n,k,u), 1 <= k <= n, u >= 0, read by rows.
6
1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1
OFFSET
1,26
COMMENTS
The number of entries per row is given by A225568.
LINKS
Christopher Hunt Gribble, C++ program
FORMULA
T(n,n,u) = A227009(n,u).
Sum_{u=1..(n-1)^2} T(n,n,u) = A034295(n).
EXAMPLE
The irregular triangle begins:
n,k\u 0 1 2 3 4 5 6 7 8 9 10 11 12 ...
1,1 1
2,1 1
2,2 1 1
3,1 1
3,2 1 1
3,3 1 1 0 0 1
4,1 1
4,2 1 1 1
4,3 1 1 1 0 1
4,4 1 1 1 1 2 0 0 0 0 1
5,1 1
5,2 1 1 1
5,3 1 1 1 0 1 1
5,4 1 1 1 1 2 1 1 0 0 1
5,5 1 1 1 1 2 1 1 1 0 1 0 0 0 ...
...
For n = 5 and k = 4 there are 2 partitions that contain 4 isolated nodes, so T(5,4,4) = 2.
Consider that each partition is composed of ones and zeros where a one represents a node with one or more links to its neighbors and a zero represents a node with no links to its neighbors. Then the 2 partitions are:
1 1 1 1 1 1 1 1 1 1
1 0 1 0 1 1 0 0 1 1
1 1 1 1 1 1 0 0 1 1
1 0 1 0 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
STATUS
approved