A361427 -id:A361427 - cp's OEIS frontend

A361424 Triangle read by rows: T(n,k) is the maximum of a certain measure of the difficulty level (see comments) for tiling an n X k rectangle with a set of integer-sided rectangular pieces, rounded down to the nearest integer.

1, 2, 2, 2, 6, 8, 4, 12, 48, 80, 4, 16, 80, 480, 1152, 8, 48, 480, 2880, 20160, 53760, 8, 53, 960, 13440, 107520

Offset: 1

Pontus von Brömssen, Mar 11 2023

The (n,k)-tiling difficulty level of a set of integer-sided rectangular pieces is defined as follows. Pieces are free to rotate, so an s X t piece and a t X s piece are equivalent. Consider a random permutation of the pieces together with a random choice of orientation of each nonsquare piece. Take one piece at a time, in the chosen order and with the chosen orientation, and try to put it with its lower left corner in the leftmost free cell in the lowest currently incomplete row of the n X k rectangle. The (n,k)-tiling difficulty level of the set of pieces is the inverse of the probability that this process results in a complete tiling of the n X k rectangle. It equals C(m; m_1, ..., m_j)*2^r/x, where m_1, ..., m_j are the number of pieces of different shapes, m is the total number of pieces, C(m; m_1, ..., m_j) is the multinomial coefficient, r is the number of nonsquare pieces, and x is the number of ways in which an n X k rectangle can be tiled with the set of pieces (where rotations and reflections are considered distinct).
The minimum possible difficulty level is 1, for example for tiling the n X k rectangle with 1 X 1 pieces only.
The difficulty level as defined here is a very crude measure of the perceived difficulty of a tiling puzzle. For example, it does not take into consideration whether a certain piece can fit in the rectangle in only one orientation. This means that the "puzzle" to put a 2 X 1 piece in a 2 X 1 rectangle, for example, has difficulty level 2, the "difficulty" being to orient the piece in the right way.
The only rectangle sizes, currently known to the author, for which the maximum difficulty level is not an integer, are 7 X 2 (difficulty level 160/3) and 13 X 2 (difficulty level 8960/3).

			Triangle begins:
  n\k|  1   2    3     4      5      6  7  8
  ---+--------------------------------------
  1  |  1
  2  |  2   2
  3  |  2   6    8
  4  |  4  12   48    80
  5  |  4  16   80   480   1152
  6  |  8  48  480  2880  20160  53760
  7  |  8  53  960 13440 107520      ?  ?
  8  | 16 120 1920 53760 483840      ?  ?  ?
For (n,k) = (7,3), the set consisting of three 1 X 2 pieces, one 1 X 5 piece, one 1 X 6 piece, and one 2 X 2 piece gives the maximum difficulty level, which equals C(6;3,1,1,1)*2^5/4 = 120*32/4 = 960 = T(7,3). The 4 in the denominator is the number of ways in which this set of pieces can tile the 7 X 3 rectangle (all 4 are equivalent under rotations and reflections).
The following table shows all sets of pieces that give the maximum (n,k)-tiling difficulty level up to (n,k) = (7,5).
      \               Number of pieces of size
  (n,k)\  1X1 | 1X2 | 1X3 | 1X4 | 1X5 | 1X6 | 1X7 | 2X2 | 2X3
  ------+-----+-----+-----+-----+-----+-----+-----+-----+----
  (1,1) |  1  |  0  |  0  |  0  |  0  |  0  |  0  |  0  |  0
  (2,1) |  0  |  1  |  0  |  0  |  0  |  0  |  0  |  0  |  0
  (2,2) |  0  |  2  |  0  |  0  |  0  |  0  |  0  |  0  |  0
  (3,1) |  1  |  1  |  0  |  0  |  0  |  0  |  0  |  0  |  0
  (3,1) |  0  |  0  |  1  |  0  |  0  |  0  |  0  |  0  |  0
  (3,2) |  1  |  1  |  1  |  0  |  0  |  0  |  0  |  0  |  0
  (3,3) |  1  |  1  |  2  |  0  |  0  |  0  |  0  |  0  |  0
  (4,1) |  0  |  2  |  0  |  0  |  0  |  0  |  0  |  0  |  0
  (4,2) |  0  |  2  |  0  |  1  |  0  |  0  |  0  |  0  |  0
  (4,2) |  0  |  1  |  2  |  0  |  0  |  0  |  0  |  0  |  0
  (4,3) |  0  |  1  |  2  |  1  |  0  |  0  |  0  |  0  |  0
  (4,4) |  0  |  1  |  2  |  2  |  0  |  0  |  0  |  0  |  0
  (5,1) |  1  |  2  |  0  |  0  |  0  |  0  |  0  |  0  |  0
  (5,1) |  0  |  1  |  1  |  0  |  0  |  0  |  0  |  0  |  0
  (5,2) |  1  |  2  |  0  |  0  |  1  |  0  |  0  |  0  |  0
  (5,2) |  0  |  3  |  0  |  1  |  0  |  0  |  0  |  0  |  0
  (5,3) |  0  |  3  |  0  |  1  |  1  |  0  |  0  |  0  |  0
  (5,3) |  0  |  1  |  3  |  1  |  0  |  0  |  0  |  0  |  0
  (5,4) |  0  |  1  |  3  |  1  |  1  |  0  |  0  |  0  |  0
  (5,5) |  1  |  0  |  8  |  0  |  0  |  0  |  0  |  0  |  0
  (6,1) |  0  |  3  |  0  |  0  |  0  |  0  |  0  |  0  |  0
  (6,2) |  0  |  1  |  2  |  1  |  0  |  0  |  0  |  0  |  0
  (6,3) |  0  |  1  |  1  |  1  |  1  |  0  |  0  |  1  |  0
  (6,4) |  0  |  1  |  1  |  1  |  1  |  1  |  0  |  1  |  0
  (6,4) |  0  |  0  |  2  |  2  |  2  |  0  |  0  |  0  |  0
  (6,5) |  0  |  0  |  2  |  2  |  2  |  1  |  0  |  0  |  0
  (6,6) |  0  |  0  |  2  |  2  |  2  |  2  |  0  |  0  |  0
  (7,1) |  1  |  3  |  0  |  0  |  0  |  0  |  0  |  0  |  0
  (7,1) |  0  |  2  |  1  |  0  |  0  |  0  |  0  |  0  |  0
  (7,2) |  0  |  1  |  4  |  0  |  0  |  0  |  0  |  0  |  0
  (7,3) |  0  |  3  |  0  |  0  |  1  |  1  |  0  |  1  |  0
  (7,3) |  0  |  1  |  1  |  1  |  0  |  1  |  0  |  0  |  1
  (7,4) |  0  |  3  |  0  |  0  |  2  |  1  |  0  |  0  |  1
  (7,4) |  0  |  0  |  3  |  2  |  1  |  1  |  0  |  0  |  0
  (7,5) |  0  |  3  |  0  |  0  |  2  |  1  |  1  |  0  |  1
  (7,5) |  0  |  0  |  3  |  2  |  1  |  1  |  1  |  0  |  0

Main diagonal: A361425.
Columns: A016116 (k = 1), A361426 (k = 2), A361427 (k = 3), A361428 (k = 4).
Cf. A360629, A361216, A361221.

T(n,1) = A016116(n).

cp's OEIS Frontend

A361424 Triangle read by rows: T(n,k) is the maximum of a certain measure of the difficulty level (see comments) for tiling an n X k rectangle with a set of integer-sided rectangular pieces, rounded down to the nearest integer.

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