A366340 -id:A366340 - 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

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

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A366341 Number of fixed (4,3)-polyominoids with n cells, allowing right-angled (or hard) connections only.

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