A075679 -id:A075679 - cp's OEIS frontend

A366766 Array read by antidiagonals, where each row is the counting sequence of a certain type of free polyominoids (see comments).

1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 3, 2, 1, 0, 1, 0, 1, 7, 5, 0, 1, 0, 1, 0, 1, 20, 16, 0, 1, 1, 0, 1, 0, 1, 60, 55, 0, 2, 1, 1, 0, 1, 0, 1, 204, 222, 0, 5, 2, 2, 1, 0, 1, 0, 1, 702, 950, 0, 12, 5, 5, 0, 1

Offset: 1

Pontus von Brömssen, Oct 22 2023

A (D,d)-polyominoid is a connected set of d-dimensional unit cubes (cells) with integer coordinates in D-dimensional space. For normal polyominoids, two cells are connected if they share a (d-1)-dimensional facet, but here we allow connections where the cells share a lower-dimensional face.
Each row is the counting sequence (by number of cells) of (D,d)-polyominoids with certain restrictions on the allowed connections between cells. Two cells have a connection of type (g,h) if they intersect in a (d-g)-dimensional unit cube and extend in d-h common dimensions. For example, d-dimensional polyominoes use connections of type (1,0), polyplets use connections of types (1,0) (edge connections) and (2,0) (corner connections), normal (3,2)-polyominoids use connections of types (1,0) ("soft" connections) and (1,1) ("hard" connections), hard polyominoids use connections of type (1,1).
Each row corresponds to a triple (D,d,C), where 1 <= d <= D and C is a set of pairs (g,h) with 1 <= g <= d and 0 <= h <= min(g, D-d). The k-th term of that row is the number of free k-celled (D,d)-polyominoids with connections of the types in C. Connections of types not in C are permitted, but the polyominoids must be connected through the specified connections only. For example, polyominoes may have cells that intersect in a point (g = 2) and hard polyominoids can have soft connections (h = 0) that are not needed to keep the polyominoids connected.
The rows are sorted first by D, then by d, and finally by a binary vector indicating which types of connections are allowed, where the connection types (g,h) are sorted lexicographically. (See table in cross-references.)
For each pair (D,d), the first row is 1, 0, 0, ..., corresponding to (D,d,{}) (no connections allowed).
The number of rows corresponding to given values of D and d is 2^((d+1)*(d+2)/2-1) if 2*d <= D and 2^((D-d+1)*(3*d-D+2)/2-1) otherwise.

			Array begins:
  n\k| 1  2  3  4  5   6    7     8      9     10      11       12
  ---+------------------------------------------------------------
   1 | 1  0  0  0  0   0    0     0      0      0       0        0
   2 | 1  1  1  1  1   1    1     1      1      1       1        1
   3 | 1  0  0  0  0   0    0     0      0      0       0        0
   4 | 1  1  1  1  1   1    1     1      1      1       1        1
   5 | 1  1  3  7 20  60  204   702   2526   9180   33989   126713
   6 | 1  2  5 16 55 222  950  4265  19591  91678  434005  2073783
   7 | 1  0  0  0  0   0    0     0      0      0       0        0
   8 | 1  1  2  5 12  35  108   369   1285   4655   17073    63600
   9 | 1  1  2  5 12  35  108   369   1285   4655   17073    63600
  10 | 1  2  5 22 94 524 3031 18770 118133 758381 4915652 32149296
  11 | 1  0  0  0  0   0    0     0      0      0       0        0
  12 | 1  1  1  1  1   1    1     1      1      1       1        1

Pontus von Brömssen, Table of n, a(n) for n = 1..210 (first 20 antidiagonals).
Pontus von Brömssen, Python programs that relate row numbers and parameter sets.
Wikipedia, Polyominoid.
Index entries for sequences related to polyominoes.

Cf. A366767 (fixed), A366768.
The following table lists some sequences that are rows of the array, together with the corresponding values of D, d, and C. Some sequences occur in more than one row. Notation used in the table:
  X: Allowed connection.
  -: Not allowed connection (but may occur "by accident" as long as it is not needed for connectedness).
  .: Not applicable for (D,d) in this row.
  !: d < D and all connections have h = 0, so these polyominoids live in d < D dimensions only.
  *: Whether a connection of type (g,h) is allowed or not is independent of h.
      |   |   | connections  |
      |   |   | g:1122233334 |
    n | D | d | h:0101201230 | sequence
  ----+---+---+--------------+---------
| 1 | 1 | * -......... | A063524
| 1 | 1 | * X......... | A000012
|!2 | 1 | * --........ | A063524
|!2 | 1 |   X-........ | A000012
| 2 | 1 |   -X........ | A361625
| 2 | 1 | * XX........ | A019988
| 2 | 2 | * -.-....... | A063524
| 2 | 2 | * X.-....... | A000105
| 2 | 2 | * -.X....... | A000105
| 2 | 2 | * X.X....... | A030222
|!3 | 1 | * --........ | A063524
|!3 | 1 |   X-........ | A000012
| 3 | 1 |   -X........ | A365654
| 3 | 1 | * XX........ | A365559
|!3 | 2 | * ----...... | A063524
|!3 | 2 |   X---...... | A000105
| 3 | 2 |   -X--...... | A365654
| 3 | 2 | * XX--...... | A075679
|!3 | 2 |   --X-...... | A000105
|!3 | 2 |   X-X-...... | A030222
| 3 | 2 |   -XX-...... | A365995
| 3 | 2 |   XXX-...... | A365997
| 3 | 2 |   ---X...... | A365999
| 3 | 2 |   X--X...... | A366001
| 3 | 2 |   -X-X...... | A366003
| 3 | 2 |   XX-X...... | A366005
| 3 | 2 | * --XX...... | A365652
| 3 | 2 |   X-XX...... | A366007
| 3 | 2 |   -XXX...... | A366009
| 3 | 2 | * XXXX...... | A365650
| 3 | 3 | * -.-..-.... | A063524
| 3 | 3 | * X.-..-.... | A038119
| 3 | 3 | * -.X..-.... | A038173
| 3 | 3 | * X.X..-.... | A268666
| 3 | 3 | * -.-..X.... | A038171
| 3 | 3 | * X.-..X.... | A363205
| 3 | 3 | * -.X..X.... | A363206
| 3 | 3 | * X.X..X.... | A272368
|!4 | 1 | * --........ | A063524
|!4 | 1 |   X-........ | A000012
| 4 | 1 |   -X........ | A366340
| 4 | 1 | * XX........ | A365561
|!4 | 2 | * -----..... | A063524
|!4 | 2 |   X----..... | A000105
| 4 | 2 |   -X---..... | A366338
| 4 | 2 | * XX---..... | A366334
|!4 | 2 |   --X--..... | A000105
|!4 | 2 |   X-X--..... | A030222
  ...
|!4 | 3 | * ----.--... | A063524
|!4 | 3 |   X---.--... | A038119
| 4 | 3 |   -X--.--... | A366340
| 4 | 3 | * XX--.--... | A366336
  ...
| 4 | 4 | * -.-..-...- | A063524
| 4 | 4 | * X.-..-...- | A068870
| 4 | 4 | * -.X..-...- | A365356
| 4 | 4 | * X.X..-...- | A365363
| 4 | 4 | * -.-..X...- | A365354
| 4 | 4 | * X.-..X...- | A365361
| 4 | 4 | * -.X..X...- | A365358
| 4 | 4 | * X.X..X...- | A365365
| 4 | 4 | * -.-..-...X | A365353
| 4 | 4 | * X.-..-...X | A365360
| 4 | 4 | * -.X..-...X | A365357
| 4 | 4 | * X.X..-...X | A365364
| 4 | 4 | * -.-..X...X | A365355
| 4 | 4 | * X.-..X...X | A365362
| 4 | 4 | * -.X..X...X | A365359
| 4 | 4 | * X.X..X...X | A365366
|!5 | 1 | * --........ | A063524
|!5 | 1 |   X-........ | A000012
| 5 | 1 |   -X........ |
| 5 | 1 | * XX........ | A365563

A365654 Number of free n-polyominoids, allowing right-angled connections only ("hard" polyominoids).

Original entry on oeis.org

1, 1, 5, 16, 90, 537, 3826, 28655, 225534

Offset: 1

Pontus von Brömssen, Sep 17 2023

Two squares are allowed to meet in a straight 180-degree connection, but the structure must be connected through right-angled ("hard") connections only. This seems to be in agreement with the definition of "hard" polyominoids in the Mireles Jasso link (the number of fixed hard hexominoids given by the "sample report" linked from that web-page agrees with A365655(6) = 22417), but differs from the definition in the Wikipedia article. The smallest example of a polyominoid that is included here but is not hard according to Wikipedia consists of two squares between (0,0,1) and (2,1,1), two between (0,0,1) and (2,0,2), and one between (1,0,0) and (1,1,1) (a "one-legged sofa", see illustration in the Mireles Jasso link). This explains why a(5) = 90, while the number of hard pentominoids is 89 according to the Wikipedia article.
Equivalently, number of n-polysticks in 3 dimensions, connected through right-angled connections.
Also, the number of face-connected polyhedral components in the square bipyramidal honeycomb up to translation, rotation, and reflection of the honeycomb. - Peter Kagey, Jun 10 2025

Pontus von Brömssen, Illustration for sizes 1..4.
Peter Kagey, Animated illustration of the a(4)=16 free polyforms on the square bipyramidal honeycomb.
Jorge Luis Mireles Jasso, The Polyominoids.
Wikipedia, Polyominoid.
Wikipedia, Polystick.
Index entries for sequences related to polyominoes.

13th and 17th row of A366766.

Cf. A075679 (polyominoids), A365559 (polysticks in 3 dimensions), A365655 (fixed).

a(9) from Pontus von Brömssen, Mar 03 2025

A365995 Number of free polyominoids with n cells, allowing flat corner-connections and right-angled edge-connections.

Original entry on oeis.org

1, 2, 9, 66, 691, 9216, 134325

Offset: 1

Pontus von Brömssen, Sep 26 2023

This sequence and the related sequences A365650-A365655 and A365996-A366010 count polyominoids (A075679) with different rules for how the cells can be connected. In these sequences, connections other than the specified ones are permitted, but the polyominoids must be connected through the specified connections only. The polyominoids counted by this sequence, for example, are allowed to have right-angled corner-connections and flat edge-connections, as long as they are not needed for the polyominoid to be connected. A connection is flat if the two neighboring cells lie in the same plane, otherwise it is right-angled.

Pontus von Brömssen, Illustration for sizes 1..3.
Wikipedia, Polyominoid.
Index entries for sequences related to polyominoes.

Cf. A365996 (fixed).
21st row of A366766.
The following table lists counting sequences for free, fixed, and one-sided polyominoids with different sets of allowed connections. "|" means flat connections and "L" means right-angled connections.
  corner-connections | edge-connections |   free  |  fixed  | 1-sided
  -------------------+------------------+---------+---------+--------
         none        |         |        | A000105 |3*A001168| A000105
         none        |          L       | A365654 | A365655 |
         none        |         |L       | A075679 | A075678 | A056846
          |          |        none      | A000105 |3*A001168| A000105
          |          |         |        | A030222 |3*A006770| A030222
          |          |          L       | A365995 | A365996 |
          |          |         |L       | A365997 | A365998 |
           L         |        none      | A365999 | A366000 |
           L         |         |        | A366001 | A366002 |
           L         |          L       | A366003 | A366004 |
           L         |         |L       | A366005 | A366006 |
          |L         |        none      | A365652 | A365653 |
          |L         |         |        | A366007 | A366008 |
          |L         |          L       | A366009 | A366010 |
          |L         |         |L       | A365650 | A365651 |

a(7) from Pontus von Brömssen, Mar 03 2025

A075678 Number of fixed (orientation matters) polyominoids (shapes made of faces of cubes) with n squares.

Original entry on oeis.org

3, 18, 158, 1611, 17811, 207395, 2505858, 31125711, 394982973, 5098498323, 66733261455, 883602795509, 11814191512434, 159283419280014, 2163058572006613

Offset: 1

Joseph Myers, Sep 24 2002

Jorge Luis Mireles Jasso, The Polyominoids (gives a(4) and a(6))

Cf. A075679 (free), A056846.

18th row of A366767.

a(13)-a(15) from John Mason, Mia Muessig, and Érika Roldán, Jul 04 2025

A366334 Number of free (4,2)-polyominoids with n cells.

Original entry on oeis.org

1, 2, 12, 95, 1267, 22349

Offset: 1

Pontus von Brömssen, Oct 07 2023

A (D,d)-polyominoid is a connected set of d-dimensional unit cubes with integer coordinates in D-dimensional space, where two cubes are connected if they share a (d-1)-dimensional facet. For example, (3,2)-polyominoids are normal polyominoids (A075679), (D,D)-polyominoids are D-dimensional polyominoes (A000105, A038119, A068870, ...), and (D,1)-polyominoids are polysticks in D dimensions (A019988, A365559, A365561, ...).

Wikipedia, Polyominoid.
Index entries for sequences related to polyominoes.

46th row of A366766.
Cf. A366335 (fixed).
Free (D,d)-polyominoids:
  D\d|    1       2       3       4
  ---+--------------------------------
   1 | A000012
   2 | A019988 A000105
   3 | A365559 A075679 A038119
   4 | A365561 A366334 A366336 A068870

A365650 Number of free n-polyominoids, allowing both corner- and edge-connections.

Original entry on oeis.org

1, 4, 36, 660, 16687

Offset: 1

Pontus von Brömssen, Sep 17 2023

These structures could be called polypletoids (or pseudo-polyominoids), because they are related to polyplets (polyominoids) as polyominoids (pseudo-polyominoes) are related to polyominoes.

Pontus von Brömssen, Illustration for sizes 1..3.
Index entries for sequences related to polyominoes.

Cf. A000105 (polyominoes), A030222 (polyplets), A075679 (polyominoids), A365651 (fixed), A365652 (corner-connections only).

30th row of A366766.

A365652 Number of free n-polyominoids, allowing corner-connections only.

Original entry on oeis.org

1, 2, 19, 293, 6590, 168753

Offset: 1

Pontus von Brömssen, Sep 17 2023

Two squares are allowed to meet in an edge, but the structure must be connected through pure corner-connections (where the two squares meet in a single point) only. There are 3 such cases of size 3; see the linked illustration.

Pontus von Brömssen, Illustration for sizes 1..3.
Index entries for sequences related to polyominoes.

Cf. A075679 (polyominoids, edge-connections only), A365650 (corner- and edge-connections), A365653 (fixed).

27th row of A366766.

A056846 Number of polyominoids containing n squares: these are 2-dimensional polyominoes in a three-dimensional grid (edge-connected squares, like the floors, ceilings and walls of a building). Mirror images are distinguished.

Original entry on oeis.org

1, 2, 11, 80, 780, 8781, 104828, 1298506, 16462696, 212457221, 2780615627, 36817036777

Offset: 1

N. J. A. Sloane, James Sellers, Aug 28 2000

R. A. Epstein, The Theory of Gambling and Statistical Logic, 1977, Academic Press, ISBN 0-12-240760-1 (pages 362-369 discuss polyforms, including Epstein's polyominoids).

M. Keller, Counting polyforms

Cf. A075679, A075678, A277565.

Corrected and extended by Joseph Myers, Sep 24 2002

A366338 Number of free (4,2)-polyominoids with n cells, allowing right-angled (or hard) connections only.

Original entry on oeis.org

1, 1, 8, 44, 509, 7091

Offset: 1

Pontus von Brömssen, Oct 07 2023

Two squares sharing an edge have a right-angled connection if they do not lie in the same plane.

Connections that are not right-angled (flat connections) may occur, but the polyominoids considered here must be connected through right-angled connections only.

Wikipedia, Polyominoid.
Index entries for sequences related to polyominoes.

45th row of A366766.

Cf. A075679, A365654, A366334, A366339 (fixed), A366340.

A366340 Number of free (4,3)-polyominoids with n cells, allowing right-angled (or hard) connections only.

Original entry on oeis.org

1, 1, 5, 19, 123, 954, 9324

Offset: 1

Pontus von Brömssen, Oct 07 2023

Two cubes sharing a face have a right-angled connection if they do not lie in the same 3-dimensional affine subspace.
Connections that are not right-angled (flat connections) may occur, but the polyominoids considered here must be connected through right-angled connections only.
Also, number of free polysticks in 4 dimensions with right-angled connections.

Wikipedia, Polyominoid.
Index entries for sequences related to polyominoes.

41st and 77th row of A366766.

Cf. A075679, A365654, A366336, A366338, A366341 (fixed).

cp's OEIS Frontend

A366766 Array read by antidiagonals, where each row is the counting sequence of a certain type of free polyominoids (see comments).

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A365654 Number of free n-polyominoids, allowing right-angled connections only ("hard" polyominoids).

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A365995 Number of free polyominoids with n cells, allowing flat corner-connections and right-angled edge-connections.

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A075678 Number of fixed (orientation matters) polyominoids (shapes made of faces of cubes) with n squares.

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A366334 Number of free (4,2)-polyominoids with n cells.

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A365650 Number of free n-polyominoids, allowing both corner- and edge-connections.

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A365652 Number of free n-polyominoids, allowing corner-connections only.

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A056846 Number of polyominoids containing n squares: these are 2-dimensional polyominoes in a three-dimensional grid (edge-connected squares, like the floors, ceilings and walls of a building). Mirror images are distinguished.

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A366338 Number of free (4,2)-polyominoids with n cells, allowing right-angled (or hard) connections only.

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A366340 Number of free (4,3)-polyominoids with n cells, allowing right-angled (or hard) connections only.

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