A361216 Triangle read by rows: T(n,k) is the maximum number of ways in which a set of integer-sided rectangular pieces can tile an n X k rectangle.

1, 1, 4, 2, 11, 56, 3, 29, 370, 5752, 4, 94, 2666, 82310, 2519124, 6, 263, 19126, 1232770, 88117873, 6126859968, 12, 968, 134902, 19119198, 2835424200

Offset: 1

Author

Pontus von Brömssen, Mar 05 2023

Keywords

• nonn
• tabl
• more

Comments

Tilings that are rotations or reflections of each other are considered distinct.

Pieces can have any combination of integer side lengths, but for the optimal sets computed so far (up to (n,k) = (7,5)), all pieces have one side of length 1.

Examples

			Triangle begins:
  n\k|  1    2       3         4            5          6  7  8
  ---+--------------------------------------------------------
  1  |  1
  2  |  1    4
  3  |  2   11      56
  4  |  3   29     370      5752
  5  |  4   94    2666     82310      2519124
  6  |  6  263   19126   1232770     88117873 6126859968
  7  | 12  968  134902  19119198   2835424200          ?  ?
  8  | 20 3416 1026667 307914196 109979838540          ?  ?  ?
A 3 X 3 square can be tiled by three 1 X 2 pieces and three 1 X 1 pieces in the following ways:
  +---+---+---+   +---+---+---+   +---+---+---+
  |   |   |   |   |   |   |   |   |   |   |   |
  +---+---+---+   +   +---+---+   +---+   +---+
  |       |   |   |   |   |   |   |   |   |   |
  +---+---+   +   +---+---+   +   +---+---+   +
  |       |   |   |       |   |   |       |   |
  +---+---+---+   +---+---+---+   +---+---+---+
.
  +---+---+---+   +---+---+---+   +---+---+---+
  |       |   |   |   |   |   |   |   |       |
  +---+---+---+   +---+---+   +   +---+---+---+
  |   |   |   |   |       |   |   |       |   |
  +---+---+   +   +---+---+---+   +---+---+---+
  |       |   |   |       |   |   |       |   |
  +---+---+---+   +---+---+---+   +---+---+---+
.
  +---+---+---+   +---+---+---+
  |       |   |   |       |   |
  +---+---+---+   +---+---+---+
  |       |   |   |   |       |
  +---+---+---+   +---+---+---+
  |       |   |   |       |   |
  +---+---+---+   +---+---+---+
The first six of these have no symmetries, so they account for 8 tilings each. The last two has a mirror symmetry, so they account for 4 tilings each. In total there are 6*8+2*4 = 56 tilings. This is the maximum for a 3 X 3 square, so T(3,3) = 56.
The following table shows the sets of pieces that give the maximum number of tilings up to (n,k) = (7,5). The solutions are unique except for (n,k) = (2,1) and (n,k) = (6,1).
      \     Number of pieces of size
  (n,k)\  1 X 1 | 1 X 2 | 1 X 3 | 1 X 4
  ------+-------+-------+-------+------
  (1,1) |   1   |   0   |   0   |   0
  (2,1) |   2   |   0   |   0   |   0
  (2,1) |   0   |   1   |   0   |   0
  (2,2) |   2   |   1   |   0   |   0
  (3,1) |   1   |   1   |   0   |   0
  (3,2) |   2   |   2   |   0   |   0
  (3,3) |   3   |   3   |   0   |   0
  (4,1) |   2   |   1   |   0   |   0
  (4,2) |   4   |   2   |   0   |   0
  (4,3) |   3   |   3   |   1   |   0
  (4,4) |   5   |   4   |   1   |   0
  (5,1) |   3   |   1   |   0   |   0
  (5,2) |   4   |   3   |   0   |   0
  (5,3) |   4   |   4   |   1   |   0
  (5,4) |   7   |   5   |   1   |   0
  (5,5) |   7   |   6   |   2   |   0
  (6,1) |   2   |   2   |   0   |   0
  (6,1) |   1   |   1   |   1   |   0
  (6,2) |   4   |   4   |   0   |   0
  (6,3) |   7   |   4   |   1   |   0
  (6,4) |   8   |   5   |   2   |   0
  (6,5) |  10   |   7   |   2   |   0
  (6,6) |  11   |   8   |   3   |   0
  (7,1) |   2   |   1   |   1   |   0
  (7,2) |   5   |   3   |   1   |   0
  (7,3) |   8   |   5   |   1   |   0
  (7,4) |  10   |   6   |   2   |   0
  (7,5) |  11   |   7   |   2   |   1
		

Crossrefs

Main diagonal: A361217.

Columns: A102462 (k = 1), A361218 (k = 2), A361219 (k = 3), A361220 (k = 4).

Cf. A360629, A361221.

Formula

T(n,1) = A102462(n).