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.

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

Author

Christopher Hunt Gribble, Jul 28 2013

Keywords

• nonn
• tabf

Comments

The number of entries per row is given by A225568.

Examples

			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
		

Links

• Christopher Hunt Gribble, Rows 1..36 for n = 1..8 and k = 1..n flattened
• Christopher Hunt Gribble, C++ program

Crossrefs

Cf. A034295, A224697, A227009, A225777, A225803, A225568.

Formula

T(n,n,u) = A227009(n,u).

Sum_{u=1..(n-1)^2} T(n,n,u) = A034295(n).