A349575 Triangle read by rows: T(w,h) (for w >= h >= 1) is the number of distinct sets of rectangles with integer sides that tile the w X h rectangle.

1, 2, 4, 3, 10, 21, 5, 22, 73, 191, 7, 44, 190, 698, 1999, 11, 91, 507, 2276, 8498, 27016, 15, 172, 1176, 6647, 29688, 116593, 390304, 22, 326, 2845, 18820, 100160, 454677, 1805924

Offset: 1

Author

Lars Blomberg, Nov 22 2021

Keywords

• nonn
• tabl
• more

Comments

The position and orientation of each rectangle in a tiling is irrelevant. Squares are allowed.

The first column is A000041 since tiling the n X 1 rectangle (ignoring the order of the tiles) is the same thing as partitioning its length n.

Examples

			Triangle begins
   1
   2   4
   3  10   21
   5  22   73   191
   7  44  190   698   1999
  11  91  507  2276   8498  27016
  15 172 1176  6647  29688 116593  390304
  22 326 2845 18820 100160 454677 1805924 ...
For w=4, h=4, 3 tiles: one 2 X 1, one 3 X 2, one 4 X 2
  aaaa
  aaaa
  bbbc
  bbbc
For w=8, h=7, 14 tiles: one 1 X 1, one 2 X 1, one 2 X 2, six 3 X 1, two 4 X 1, one 4 X 3, one 5 X 1, one 6 X 1
  aaaagggl
  aaaaffkl
  aaaaffkl
  bbbbbbkn
  cccccjjj
  ddddhhhm
  eeeeiiim
		

Crossrefs

Cf. A000041, A348133.