cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

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.

Original entry on oeis.org

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

Views

Author

Pontus von Brömssen, Mar 05 2023

Keywords

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).

A362258 Triangle read by rows: T(n,k) is the maximum number of ways in which a set of integer-sided squares can tile an n X k rectangle, up to rotations and reflections, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 4, 1, 1, 2, 4, 13, 20, 1, 1, 4, 8, 33, 125, 277, 1, 1, 6, 12, 72, 403, 2505, 7855, 1, 1, 9, 22, 204, 1438, 12069, 101587, 487662
Offset: 0

Views

Author

Pontus von Brömssen, Apr 15 2023

Keywords

Examples

			Triangle begins:
  n\k| 0  1  2  3   4    5     6      7      8
  ---+----------------------------------------
  0  | 1
  1  | 1  1
  2  | 1  1  1
  3  | 1  1  1  1
  4  | 1  1  2  2   4
  5  | 1  1  2  4  13   20
  6  | 1  1  4  8  33  125   277
  7  | 1  1  6 12  72  403  2505   7855
  8  | 1  1  9 22 204 1438 12069 101587 487662
See A362142 for an illustration of T(5,4) = 13.
The following table shows which sets of squares can tile the n X k rectangle in T(n,k) ways. A list x_1, ..., x_j represents a set of x_1 squares of side 1, ..., x_j squares of side j. When there are multiple solutions they are shown on separate lines. For (n,k) = (4,3), for example, the maximum number T(4,3) = 2 of tilings is obtained both for the set of 8 squares of side 1 and 1 square of side 2, and for the set of 4 squares of side 1 and 2 squares of side 2.
  n\k| 1   2      3     4     5     6      7       8
  ---+------------------------------------------------
  1  | 1
  2  | 2  4
     |    0,1
  3  | 3  6     9
     |    2,1   5,1
     |          0,0,1
  4  | 4  4,1   8,1    8,2
     |          4,2
  5  | 5  6,1   7,2   12,2  13,3
     |    2,2
  6  | 6  4,2  10,2   12,3  14,4  20,4
  7  | 7  6,2  13,2   12,4  19,4  22,5  25,6
  8  | 8  8,2  12,3   16,4  20,5  24,6  23,6,1  27,7,1

Crossrefs

Main diagonal: A362259.
Columns: A000012 (k = 0,1), A362260 (k = 2), A362261 (k = 3), A362262 (k = 4), A362263 (k = 5).
Cf. A227690, A361221 (rectangular pieces), A362142.

Formula

T(n,k) >= A362142(n,k)/4 if n != k.
T(n,n) >= A362142(n,n)/8.

A361222 Maximum number of ways in which a set of integer-sided rectangular pieces can tile an n X n square, up to rotations and reflections.

Original entry on oeis.org

1, 1, 8, 719, 315107
Offset: 1

Views

Author

Pontus von Brömssen, Mar 05 2023

Keywords

Crossrefs

Main diagonal of A361221.
Cf. A361217, A360630.

A361224 Maximum number of ways in which a set of integer-sided rectangular pieces can tile an n X 2 rectangle, up to rotations and reflections.

Original entry on oeis.org

1, 1, 5, 12, 31, 86, 242, 854, 2888, 10478, 34264, 120347
Offset: 1

Views

Author

Pontus von Brömssen, Mar 05 2023

Keywords

Examples

			A 4 X 2 rectangle can be tiled by two 1 X 2 pieces and four 1 X 1 pieces in the following 12 ways:
  +---+---+   +---+---+   +---+---+   +---+---+   +---+---+   +---+---+
  |   |   |   |   |   |   |   |   |   |   |   |   |       |   |   |   |
  +---+---+   +---+---+   +---+---+   +   +---+   +---+---+   +---+---+
  |   |   |   |   |   |   |       |   |   |   |   |   |   |   |   |   |
  +---+---+   +   +---+   +---+---+   +---+---+   +---+---+   +---+---+
  |       |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
  +---+---+   +---+---+   +---+---+   +---+---+   +---+---+   +   +   +
  |       |   |       |   |       |   |       |   |       |   |   |   |
  +---+---+   +---+---+   +---+---+   +---+---+   +---+---+   +---+---+
.
  +---+---+   +---+---+   +---+---+   +---+---+   +---+---+   +---+---+
  |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
  +---+---+   +---+---+   +   +---+   +---+   +   +---+---+   +---+---+
  |   |   |   |       |   |   |   |   |   |   |   |       |   |   |   |
  +---+   +   +---+---+   +---+---+   +---+---+   +---+---+   +   +   +
  |   |   |   |   |   |   |   |   |   |   |   |   |       |   |   |   |
  +   +---+   +   +---+   +   +---+   +   +---+   +---+---+   +---+---+
  |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
  +---+---+   +---+---+   +---+---+   +---+---+   +---+---+   +---+---+
This is the maximum for a 4 X 2 rectangle, so a(4) = 12.
The following table shows the sets of pieces that give the maximum number of tilings for n <= 12. The solutions are unique except for n <= 2.
    \     Number of pieces of size
   n \  1 X 1 | 1 X 2 | 1 X 3 | 2 X 2
  ----+-------+-------+-------+------
   1  |   2   |   0   |   0   |   0
   1  |   0   |   1   |   0   |   0
   2  |   4   |   0   |   0   |   0
   2  |   2   |   1   |   0   |   0
   2  |   0   |   2   |   0   |   0
   2  |   0   |   0   |   0   |   1
   3  |   2   |   2   |   0   |   0
   4  |   4   |   2   |   0   |   0
   5  |   4   |   3   |   0   |   0
   6  |   4   |   4   |   0   |   0
   7  |   5   |   3   |   1   |   0
   8  |   5   |   4   |   1   |   0
   9  |   7   |   4   |   1   |   0
  10  |   7   |   5   |   1   |   0
  11  |   7   |   6   |   1   |   0
  12  |   9   |   6   |   1   |   0
It seems that all optimal solutions for A361218 are also optimal here, but for n = 2 there are other optimal solutions.

Crossrefs

Second column of A361221.
Cf. A361218, A360631.

A361225 Maximum number of ways in which a set of integer-sided rectangular pieces can tile an n X 3 rectangle, up to rotations and reflections.

Original entry on oeis.org

1, 5, 8, 95, 682, 4801, 33807
Offset: 1

Views

Author

Pontus von Brömssen, Mar 05 2023

Keywords

Examples

			The following table shows the sets of pieces that give the maximum number of tilings for n <= 7. The solutions are unique except for n = 1 and n = 3.
   \ Number of pieces of size
  n \  1 X 1 | 1 X 2 | 1 X 3
  ---+-------+-------+------
  1  |   3   |   0   |   0
  1  |   1   |   1   |   0
  1  |   0   |   0   |   1
  2  |   2   |   2   |   0
  3  |   3   |   3   |   0
  3  |   2   |   2   |   1
  4  |   3   |   3   |   1
  5  |   4   |   4   |   1
  6  |   7   |   4   |   1
  7  |   8   |   5   |   1
It seems that all optimal solutions for A361219 are also optimal here, but for n = 1 and n = 3 there are other optimal solutions.

Crossrefs

Third column of A361221.
Cf. A361219, A360632.

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.

Original entry on oeis.org

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

Views

Author

Pontus von Brömssen, Mar 11 2023

Keywords

Comments

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).

Examples

			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

Crossrefs

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

Formula

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

A361223 Maximum number of inequivalent permutations of a partition of n, where two permutations are equivalent if they are reversals of each other.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 6, 10, 16, 30, 54, 84, 140, 252, 420, 756, 1260, 2520, 4620, 7920, 13860, 27720, 51480, 90120, 180180, 337890, 600600, 1081080, 2042040, 3675672, 6348888, 12252240, 23279256, 42325920, 77597520, 148140720, 271591320, 480507720, 892371480
Offset: 1

Views

Author

Pontus von Brömssen, Mar 05 2023

Keywords

Examples

			For n = 5, the 7 partitions have the following permutations (~ means equivalence under reversal):
  permutations         | number of inequivalent permutations
  ---------------------+------------------------------------
          5            |   1
        41~14          |   1
        32~23          |   1
    311~113, 131       |   2
    221~122, 212       |   2
  2111~1112, 1211~1121 |   2
        11111          |   1
The maximum number of inequivalent permutations is 2 (for the partitions 311, 221, and 2111), so a(5) = 2.

Links

Crossrefs

First column of A361221.
Cf. A102462.
Showing 1-7 of 7 results.

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