A361523 Triangle read by rows: T(n,k) is the number of ways of dividing an n X k rectangle into integer-sided rectangles, up to rotations and reflections.

1, 1, 1, 1, 2, 4, 1, 3, 17, 54, 1, 6, 61, 892, 9235, 1, 10, 220, 8159, 406653, 10538496

Offset: 0

Author

Pontus von Brömssen, Mar 15 2023

Keywords

• nonn
• tabl
• more

Examples

			Triangle begins:
  n\k| 0  1   2    3      4        5
  ---+------------------------------
  0  | 1
  1  | 1  1
  2  | 1  2   4
  3  | 1  3  17   54
  4  | 1  6  61  892   9235
  5  | 1 10 220 8159 406653 10538496
The 3 X 2 rectangle can be divided in T(3,2) = 17 inequivalent ways:
  +---+---+   +---+---+   +---+---+   +---+---+   +---+---+   +---+---+
  |       |   |       |   |   |   |   |   |   |   |   |   |   |   |   |
  +       +   +---+---+   +---+---+   +   +   +   +   +---+   +   +---+
  |       |   |       |   |       |   |   |   |   |   |   |   |   |   |
  +       +   +       +   +       +   +   +   +   +   +   +   +   +---+
  |       |   |       |   |       |   |   |   |   |   |   |   |   |   |
  +---+---+   +---+---+   +---+---+   +---+---+   +---+---+   +---+---+
.
  +---+---+   +---+---+   +---+---+   +---+---+   +---+---+   +---+---+
  |       |   |   |   |   |   |   |   |   |   |   |       |   |   |   |
  +---+---+   +   +   +   +---+---+   +   +---+   +---+---+   +---+---+
  |       |   |   |   |   |       |   |   |   |   |   |   |   |   |   |
  +---+---+   +---+---+   +---+---+   +---+---+   +---+---+   +   +   +
  |       |   |       |   |       |   |       |   |       |   |   |   |
  +---+---+   +---+---+   +---+---+   +---+---+   +---+---+   +---+---+
.
  +---+---+   +---+---+   +---+---+   +---+---+   +---+---+
  |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
  +---+   +   +---+---+   +---+---+   +---+---+   +---+---+
  |   |   |   |   |   |   |   |   |   |       |   |   |   |
  +   +---+   +---+---+   +   +---+   +---+---+   +---+---+
  |   |   |   |       |   |   |   |   |   |   |   |   |   |
  +---+---+   +---+---+   +---+---+   +---+---+   +---+---+
		

Crossrefs

Main diagonal: A361524.

Columns: A000012 (k = 0), A005418 (k = 1), A347825 (k = 2; with an exception for n = 2), A361525 (k = 3), A361526 (k = 4).

Cf. A116694 (rotations and reflections are considered distinct), A227690 (square pieces), A360629.

Formula

T(n,k) >= A116694(n,k)/4 if n != k.

T(n,n) >= A116694(n,n)/8.