A225803 -id:A225803 - cp's OEIS frontend

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

Christopher Hunt Gribble, Jul 28 2013

The number of entries per row is given by A225568.

			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

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

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

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

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

A225777 Number T(n,k,u) of distinct tilings of an n X k rectangle using integer-sided square tiles containing u nodes that are unconnected to any of their neighbors; irregular triangle T(n,k,u), 1 <= k <= n, u >= 0, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 4, 0, 0, 1, 1, 1, 3, 1, 1, 6, 4, 0, 2, 1, 9, 16, 8, 5, 0, 0, 0, 0, 1, 1, 1, 4, 3, 1, 8, 12, 0, 3, 4, 1, 12, 37, 34, 15, 12, 4, 0, 0, 2, 1, 16, 78, 140, 88, 44, 68, 32, 0, 4, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Christopher Hunt Gribble, Jul 26 2013

The number of entries per row is given by A225568.

			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   2
3,3   1   4   0   0   1
4,1   1
4,2   1   3   1
4,3   1   6   4   0   2
4,4   1   9  16   8   5   0   0   0   0   1
5,1   1
5,2   1   4   3
5,3   1   8  12   0   3   4
5,4   1  12  37  34  15  12   4   0   0   2
5,5   1  16  78 140  88  44  68  32   0   4   0   0   0 ...
...
For n = 4, k = 3, there are 4 tilings that contain 2 isolated nodes, so T(4,3,2) = 4. A 2 X 2 square contains 1 isolated node.  Consider that each tiling 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 4 tilings are:
   1 1 1 1    1 1 1 1    1 1 1 1    1 1 1 1
   1 0 1 1    1 0 1 1    1 1 0 1    1 1 0 1
   1 1 1 1    1 1 1 1    1 1 1 1    1 1 1 1
   1 0 1 1    1 1 0 1    1 0 1 1    1 1 0 1
   1 1 1 1    1 1 1 1    1 1 1 1    1 1 1 1

Christopher Hunt Gribble, Rows 1..36 for n=2..8 and k=1..n flattened
Christopher Hunt Gribble, C++ program

Cf. A045846, A219924, A226997, A225542, A225803, A225568.

T(n,k,0) = 1, T(n,k,1) = (n-1)(k-1), T(n,k,2) = (n^2(k-1) - n(2k^2+5k-13) + (k^2+13k-24))/2.

Sum_{u=1..(n-1)^2} T(n,n,u) = A045846(n).

cp's OEIS Frontend

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.

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