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.

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

Original entry on oeis.org

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

Views

Author

Pontus von Brömssen, Oct 22 2023

Keywords

Comments

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.

Examples

			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

Links

Crossrefs

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 | 1 | * -......... | A063524
2 | 1 | 1 | * X......... | A000012
3 |!2 | 1 | * --........ | A063524
4 |!2 | 1 | X-........ | A000012
5 | 2 | 1 | -X........ | A361625
6 | 2 | 1 | * XX........ | A019988
7 | 2 | 2 | * -.-....... | A063524
8 | 2 | 2 | * X.-....... | A000105
9 | 2 | 2 | * -.X....... | A000105
10 | 2 | 2 | * X.X....... | A030222
11 |!3 | 1 | * --........ | A063524
12 |!3 | 1 | X-........ | A000012
13 | 3 | 1 | -X........ | A365654
14 | 3 | 1 | * XX........ | A365559
15 |!3 | 2 | * ----...... | A063524
16 |!3 | 2 | X---...... | A000105
17 | 3 | 2 | -X--...... | A365654
18 | 3 | 2 | * XX--...... | A075679
19 |!3 | 2 | --X-...... | A000105
20 |!3 | 2 | X-X-...... | A030222
21 | 3 | 2 | -XX-...... | A365995
22 | 3 | 2 | XXX-...... | A365997
23 | 3 | 2 | ---X...... | A365999
24 | 3 | 2 | X--X...... | A366001
25 | 3 | 2 | -X-X...... | A366003
26 | 3 | 2 | XX-X...... | A366005
27 | 3 | 2 | * --XX...... | A365652
28 | 3 | 2 | X-XX...... | A366007
29 | 3 | 2 | -XXX...... | A366009
30 | 3 | 2 | * XXXX...... | A365650
31 | 3 | 3 | * -.-..-.... | A063524
32 | 3 | 3 | * X.-..-.... | A038119
33 | 3 | 3 | * -.X..-.... | A038173
34 | 3 | 3 | * X.X..-.... | A268666
35 | 3 | 3 | * -.-..X.... | A038171
36 | 3 | 3 | * X.-..X.... | A363205
37 | 3 | 3 | * -.X..X.... | A363206
38 | 3 | 3 | * X.X..X.... | A272368
39 |!4 | 1 | * --........ | A063524
40 |!4 | 1 | X-........ | A000012
41 | 4 | 1 | -X........ | A366340
42 | 4 | 1 | * XX........ | A365561
43 |!4 | 2 | * -----..... | A063524
44 |!4 | 2 | X----..... | A000105
45 | 4 | 2 | -X---..... | A366338
46 | 4 | 2 | * XX---..... | A366334
47 |!4 | 2 | --X--..... | A000105
48 |!4 | 2 | X-X--..... | A030222
...
75 |!4 | 3 | * ----.--... | A063524
76 |!4 | 3 | X---.--... | A038119
77 | 4 | 3 | -X--.--... | A366340
78 | 4 | 3 | * XX--.--... | A366336
...
139 | 4 | 4 | * -.-..-...- | A063524
140 | 4 | 4 | * X.-..-...- | A068870
141 | 4 | 4 | * -.X..-...- | A365356
142 | 4 | 4 | * X.X..-...- | A365363
143 | 4 | 4 | * -.-..X...- | A365354
144 | 4 | 4 | * X.-..X...- | A365361
145 | 4 | 4 | * -.X..X...- | A365358
146 | 4 | 4 | * X.X..X...- | A365365
147 | 4 | 4 | * -.-..-...X | A365353
148 | 4 | 4 | * X.-..-...X | A365360
149 | 4 | 4 | * -.X..-...X | A365357
150 | 4 | 4 | * X.X..-...X | A365364
151 | 4 | 4 | * -.-..X...X | A365355
152 | 4 | 4 | * X.-..X...X | A365362
153 | 4 | 4 | * -.X..X...X | A365359
154 | 4 | 4 | * X.X..X...X | A365366
155 |!5 | 1 | * --........ | A063524
156 |!5 | 1 | X-........ | A000012
157 | 5 | 1 | -X........ |
158 | 5 | 1 | * XX........ | A365563

A366767 Array read by antidiagonals, where each row is the counting sequence of a certain type of fixed polyominoids.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 0, 2, 0, 1, 0, 2, 2, 0, 1, 0, 2, 4, 2, 0, 1, 0, 2, 12, 6, 1, 0, 1, 0, 2, 38, 22, 0, 1, 0, 1, 0, 2, 126, 88, 0, 2, 1, 0, 1, 0, 2, 432, 372, 0, 6, 2, 1, 0, 1, 0, 2, 1520, 1628, 0, 19, 6, 4, 3, 0, 1, 0, 2, 5450, 7312, 0, 63, 19, 20, 0, 3
Offset: 1

Views

Author

Pontus von Brömssen, Oct 22 2023

Keywords

Comments

See A366766 (corresponding array for free polyominoids) for details.

Examples

			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 | 2  0  0   0   0    0     0      0      0       0        0         0
   4 | 2  2  2   2   2    2     2      2      2       2        2         2
   5 | 2  4 12  38 126  432  1520   5450  19820   72892   270536   1011722
   6 | 2  6 22  88 372 1628  7312  33466 155446  730534  3466170  16576874
   7 | 1  0  0   0   0    0     0      0      0       0        0         0
   8 | 1  2  6  19  63  216   760   2725   9910   36446   135268    505861
   9 | 1  2  6  19  63  216   760   2725   9910   36446   135268    505861
  10 | 1  4 20 110 638 3832 23592 147941 940982 6053180 39299408 257105146
  11 | 3  0  0   0   0    0     0      0      0       0        0         0
  12 | 3  3  3   3   3    3     3      3      3       3        3         3

Links

Crossrefs

Cf. A366766 (free), A366768.
The following table lists some sequences that are rows of the array, together with the corresponding values of D, d, and C (see A366766). 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:112223 |
n | D | d | h:010120 | sequence
----+---+---+-------------+----------
1 | 1 | 1 | * -..... | A063524
2 | 1 | 1 | * X..... | A000012
3 |!2 | 1 | * --.... | 2*A063524
4 |!2 | 1 | X-.... | 2*A000012
5 | 2 | 1 | -X.... | 2*A001168
6 | 2 | 1 | * XX.... | A096267
7 | 2 | 2 | * -.-... | A063524
8 | 2 | 2 | * X.-... | A001168
9 | 2 | 2 | * -.X... | A001168
10 | 2 | 2 | * X.X... | A006770
11 |!3 | 1 | * --.... | 3*A063524
12 |!3 | 1 | X-.... | 3*A000012
13 | 3 | 1 | -X.... | A365655
14 | 3 | 1 | * XX.... | A365560
15 |!3 | 2 | * ----.. | 3*A063524
16 |!3 | 2 | X---.. | 3*A001168
17 | 3 | 2 | -X--.. | A365655
18 | 3 | 2 | * XX--.. | A075678
19 |!3 | 2 | --X-.. | 3*A001168
20 |!3 | 2 | X-X-.. | 3*A006770
21 | 3 | 2 | -XX-.. | A365996
22 | 3 | 2 | XXX-.. | A365998
23 | 3 | 2 | ---X.. | A366000
24 | 3 | 2 | X--X.. | A366002
25 | 3 | 2 | -X-X.. | A366004
26 | 3 | 2 | XX-X.. | A366006
27 | 3 | 2 | * --XX.. | A365653
28 | 3 | 2 | X-XX.. | A366008
29 | 3 | 2 | -XXX.. | A366010
30 | 3 | 2 | * XXXX.. | A365651
31 | 3 | 3 | * -.-..- | A063524
32 | 3 | 3 | * X.-..- | A001931
33 | 3 | 3 | * -.X..- | A039742
34 | 3 | 3 | * X.X..- |
35 | 3 | 3 | * -.-..X | A039741
36 | 3 | 3 | * X.-..X |
37 | 3 | 3 | * -.X..X |
38 | 3 | 3 | * X.X..X |
39 |!4 | 1 | * --.... | 4*A063524
40 |!4 | 1 | X-.... | 4*A000012
41 | 4 | 1 | -X.... | A366341
42 | 4 | 1 | * XX.... | A365562
43 |!4 | 2 | * -----. | 6*A063524
44 |!4 | 2 | X----. | 6*A001168
45 | 4 | 2 | -X---. | A366339
46 | 4 | 2 | * XX---. | A366335
47 |!4 | 2 | --X--. | 6*A001168
48 |!4 | 2 | X-X--. | 6*A006770

A383735 Array read by antidiagonals, where each row is the cluster series for percolation on the cells of a certain type of polyominoids.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 2, 0, 1, 0, 2, 0, 2, 1, 0, 2, 0, 2, 4, 1, 0, 2, 0, 2, 12, 6, 1, 0, 2, 0, 2, 24, 18, 0, 1, 0, 2, 0, 2, 52, 48, 0, 4, 1, 0, 2, 0, 2, 108, 126, 0, 12, 4, 1, 0, 2, 0, 2, 224, 300, 0, 24, 12, 8, 1, 0, 2, 0, 2, 412, 762, 0, 52, 24, 32, 0, 1
Offset: 1

Views

Author

Pontus von Brömssen, May 10 2025

Keywords

Comments

T(n,k) is the coefficient of p^(k+1), k >= 0, in the power series expansion of the expected finite size of the cluster containing a given cell for percolation with probability p on the polyominoid cells corresponding to row n of A366766. If the given cell is not open, its cluster is empty. Equivalently, T(n,k) can be taken to be the coefficient of p^k if we condition on the event that the given cell is open.
See A366766 for details on how the polyominoids are specified and on the ordering of the rows.

Examples

			Array begins:
  n\k| 0  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       0
   2 | 1  2  2   2   2    2    2    2     2     2      2      2       2
   3 | 1  0  0   0   0    0    0    0     0     0      0      0       0
   4 | 1  2  2   2   2    2    2    2     2     2      2      2       2
   5 | 1  4 12  24  52  108  224  412   844  1528   3152   5036   11984
   6 | 1  6 18  48 126  300  762 1668  4216  8668  21988  43058  110832
   7 | 1  0  0   0   0    0    0    0     0     0      0      0       0
   8 | 1  4 12  24  52  108  224  412   844  1528   3152   5036   11984
   9 | 1  4 12  24  52  108  224  412   844  1528   3152   5036   11984
  10 | 1  8 32 108 348 1068 3180 9216 26452 73708 206872 563200 1555460
  11 | 1  0  0   0   0    0    0    0     0     0      0      0       0
  12 | 1  2  2   2   2    2    2    2     2     2      2      2       2

Links

Crossrefs

Cf. A366766, A366767, A366768.
Rows include:
n | sequence for row n
---+-------------------
1 | A000007
2 | A040000
3 | A000007
4 | A040000
5 | A003203
6 | A003198
7 | A000007
8 | A003203
9 | A003203
10 | A003201
11 | A000007
12 | A040000
13 | A383737
14 | A003207
15 | A000007
16 | A003203
17 | A383737
18 | A383736
19 | A003203
20 | A003201
...
31 | A000007
32 | A003211
33 | A003209
34 | A036396
35 | A003210
...
38 | A036402
39 | A000007
40 | A040000
...
43 | A000007
44 | A003203
...
47 | A003203
48 | A003201

Formula

T(n,k) = [p^k] Sum_P m^2*p^(m-1)*(1-p)^j / binomial(D,d) = Sum_P m^2*(-1)^(k-m+1)*binomial(j,k-m+1) / binomial(D,d), where the sum is over all fixed polyominoids P (corresponding to row n of A366766), m is the number of cells of P, and j is the number of cells that are not in P but are adjacent to a cell in P; d is the dimension of the cells and D is the dimension of the ambient space. It is sufficient to take the sums over those P that have at most k+1 cells.

A383736 Cluster series for percolation on polyominoid cells.

Original entry on oeis.org

1, 12, 92, 604, 3732, 22766, 136564
Offset: 0

Views

Author

Pontus von Brömssen, May 10 2025

Keywords

Comments

The coordination sequence for polyominoid cells (row 18 of A366768) is A005914, except that the term for distance 1 should be 12.

Links

Crossrefs

Row 18 of A383735.
Cf. A005914, A075678, A075679, A366768, A383737.
Showing 1-4 of 4 results.

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