A361221 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, up to rotations and reflections.

1, 1, 1, 1, 5, 8, 2, 12, 95, 719, 2, 31, 682, 20600, 315107

Offset

1,5

Links

Table of n, a(n) for n=1..15.

Example

Triangle begins:
  n\k|  1  2    3     4      5
  ---+------------------------
  1  |  1
  2  |  1  1
  3  |  1  5    8
  4  |  2 12   95   719
  5  |  2 31  682 20600 315107
A 3 X 3 square can be tiled by one 1 X 3 piece, two 1 X 2 pieces and two 1 X 1 pieces in the following 8 ways:
  +---+---+---+   +---+---+---+   +---+---+---+
  |   |   |   |   |       |   |   |   |       |
  +---+---+   +   +---+---+---+   +---+---+---+
  |       |   |   |       |   |   |       |   |
  +---+---+---+   +---+---+---+   +---+---+---+
  |           |   |           |   |           |
  +---+---+---+   +---+---+---+   +---+---+---+
.
  +---+---+---+   +---+---+---+   +---+---+---+
  |   |   |   |   |   |   |   |   |   |       |
  +   +   +---+   +   +---+   +   +   +---+---+
  |   |   |   |   |   |   |   |   |   |   |   |
  +---+---+---+   +---+---+---+   +---+---+---+
  |           |   |           |   |           |
  +---+---+---+   +---+---+---+   +---+---+---+
.
  +---+---+---+   +---+---+---+
  |       |   |   |   |       |
  +---+---+---+   +---+---+---+
  |           |   |           |
  +---+---+---+   +---+---+---+
  |       |   |   |       |   |
  +---+---+---+   +---+---+---+
This is the maximum for a 3 X 3 square, so T(3,3) = 8. There is one other set of pieces that also can tile the 3 X 3 square in 8 ways: three 1 X 2 pieces and three 1 X 1 pieces (see illustration in A361216).
The following table shows all sets of pieces that give the maximum number of tilings for 1 <= k <= n <= 5:
      \     Number of pieces of size
  (n,k)\  1 X 1 | 1 X 2 | 1 X 3 | 2 X 2
  ------+-------+-------+-------+------
  (1,1) |   1   |   0   |   0   |   0
  (2,1) |   2   |   0   |   0   |   0
  (2,1) |   0   |   1   |   0   |   0
  (2,2) |   4   |   0   |   0   |   0
  (2,2) |   2   |   1   |   0   |   0
  (2,2) |   0   |   2   |   0   |   0
  (2,2) |   0   |   0   |   0   |   1
  (3,1) |   3   |   0   |   0   |   0
  (3,1) |   1   |   1   |   0   |   0
  (3,1) |   0   |   0   |   1   |   0
  (3,2) |   2   |   2   |   0   |   0
  (3,3) |   3   |   3   |   0   |   0
  (3,3) |   2   |   2   |   1   |   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
  (5,1) |   2   |   0   |   1   |   0
  (5,1) |   1   |   2   |   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
It seems that all optimal solutions for A361216 are also optimal here, but occasionally there are other optimal solutions, e.g. for n = k = 3.

Crossrefs

Main diagonal: A361222.

Columns: A361223 (k = 1), A361224 (k = 2), A361225 (k = 3).

Cf. A360629, A361216.

Sequence in context: A198844 A156035 A284697 * A319261 A010489 A196567

Adjacent sequences: A361218 A361219 A361220 * A361222 A361223 A361224

Keyword

nonn,tabl,more

Author

Pontus von Brömssen, Mar 05 2023

Status

approved