A227998 Number T(n,k,s) of partitions of an n X k rectangle into s integer-sided squares, considering only the list of parts; irregular triangle T(n,k,s), 1 <= k <= n, s >= 1, read by rows.

1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0

Offset

1

Comments

The number of entries per row is n*k.

Links

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

Christopher Hunt Gribble, C++ program

Formula

T(n,n,s) = A226912(n,s).

Sum_{s=1..n*k} T(n,k,s) = A224697(n,k), 1 <= k <= n.

Example

T(6,4,6) = 2 because there are 2 partitions of a 6 X 4 rectangle into integer-sided squares with exactly 6 parts:
   (6 2 X 2 squares) and
   (4 1 X 1 squares, 1 2 X 2 square, 1 4 X 4 square).
The irregular triangle starts:
n,k  Number of Square Parts s
     1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 ...
1,1  1
2,1  0  1
2,2  1  0  0  1
3,1  0  0  1
3,2  0  0  1  0  0  1
3,3  1  0  0  0  0  1  0  0  1
4,1  0  0  0  1
4,2  0  1  0  0  1  0  0  1
4,3  0  0  0  1  0  1  0  0  1  0  0  1
4,4  1  0  0  1  0  0  1  1  0  1  0  0  1  0  0  1
5,1  0  0  0  0  1
5,2  0  0  0  1  0  0  1  0  0  1
5,3  0  0  0  1  0  0  1  0  1  0  0  1  0  0  1
5,4  0  0  0  0  1  1  0  1  1  0  1  1  0  1  0  0  1  0 ...
5,5  1  0  0  0  0  0  0  1  0  1  1  0  1  1  0  1  1  0 ...
6,1  0  0  0  0  0  1
6,2  0  0  1  0  0  1  0  0  1  0  0  1
6,3  0  1  0  0  0  0  1  0  1  1  0  1  0  0  1  0  0  1
6,4  0  0  1  0  0  2  0  1  2  1  0  1  1  0  1  1  0  1 ...
6,5  0  0  0  1  1  0  1  1  1  1  2  1  1  2  1  0  1  1 ...
6,6  1  0  0  1  0  1  0  0  3  0  1  4  1  1  2  1  1  2 ...

Crossrefs

Cf. A034295, A226912.

Sequence in context: A280816 A265246 A138709 * A266514 A082960 A320815

Adjacent sequences: A227995 A227996 A227997 * A227999 A228000 A228001

Keyword

nonn,tabf

Author

Christopher Hunt Gribble, Aug 06 2013

Status

approved