Polyomino sequence

[Victor Miller]

This discussion on Mastodon 

https://mathstodon.xyz/@jsiehler/114744887537037820 should be able to generate a new sequence: For each n let a(n) be the number of symmetric connected regions that can be built out of n+1 n-ominoes.

[Arthur O'Dwyer]

On Fri, Aug 15, 2025, 8:13 AM Victor Miller <victor...@gmail.com> wrote:

This discussion on Mastodon 

https://mathstodon.xyz/@jsiehler/114744887537037820 should be able to generate a new sequence: For each n let a(n) be the number of symmetric connected regions that can be built out of n+1 n-ominoes.

Surely you mean "that can be built out of a complete set of (free) n-ominoes."

For example there are 5 4-ominoes, but there are only 2 3-ominoes and there are 12 5-ominoes.

- Axxxxxx(n) = 1, 1, 9, ... is the number of connected free regions that can be built out of the complete set of n-ominoes.

IMO it's not worth thinking about "symmetric" regions because there are so many things that could mean. The OP on Mastodon seems to have been thinking only of bilateral symmetry where the axis of symmetry is horizontal. It's not obvious to me why you couldn't build a region out of the five tetrominoes that would be bilaterally symmetric about a diagonal axis. (That's the only kind of bilateral symmetry you can build with the 2 trominoes, for example.)  And what about rotational symmetries?

Pencil and paper suggests that there are

- 6 connected free regions buildable out of 2 free I-trominoes — 3 of which are bilaterally symmetric;

- 15 connected free regions buildable out of 2 free V-trominoes — 4 of which are bilaterally symmetric;

- 9 connected free regions buildable out of a complete set of the 2 trominoes — 1 of which is bilaterally symmetric;

- 28 connected free regions buildable out of 2 free trominoes — 7 of which are bilaterally symmetric.

A000105(n) is the number of connected free regions that can be built out of n free monominoes (m=1). These are the "free polyominoes".

A056785(n) is the number of connected free regions that can be built out of n free dominoes (m=2). These are the "free polydominoes".

I think it would be most satisfying to submit a sequence that continues to generalize (A000105, A056785, ...). That is, either of:

- Axxxxxx(n) = 2, 28, ... is the number of connected free regions that can be built out of n (not necessarily distinct) free trominoes (m=3). These are the "free polytrominoes".

- Axxxxxx(n) = 1, 6, ... is the number of connected free regions that can be built out of n (not necessarily distinct) free straight trominoes. One might christen these the "free polybars of order m=3".

(I just made up the term "polybar". I don't see either the term or the concept of poly-straight-polyominoes listed on Miroslav Vicher's page nor Michael Keller's page. I do see poly-square-tetrominoes termed "polynars" on Abaroth's page.)

The difficulty, I guess, is that to compute the 3rd term you're already counting distinct 9-cell regions; the 4th term counts 12-cell regions; etc., which seems unwieldy to handle except by (maybe even by) computer.

I'm not even very sure that I got my numbers above correct. In the process of composing this email I found two animals I'd missed. :)

–Arthur

[Arthur O'Dwyer]

On Fri, Aug 15, 2025 at 9:36 AM Arthur O'Dwyer <arthur....@gmail.com> wrote:

On Fri, Aug 15, 2025, 8:13 AM Victor Miller <victor...@gmail.com> wrote:

This discussion on Mastodon 

https://mathstodon.xyz/@jsiehler/114744887537037820 should be able to generate a new sequence: For each n let a(n) be the number of symmetric connected regions that can be built out of n+1 n-ominoes.

Surely you mean "that can be built out of a complete set of (free) n-ominoes."

For example there are 5 4-ominoes, but there are only 2 3-ominoes and there are 12 5-ominoes.

- Axxxxxx(n) = 1, 1, 9, ... is the number of connected free regions that can be built out of the complete set of n-ominoes.

My initial pencil-and-paper count was wrong: the third term is "11", not "9".

 

[...]

A000105(n) is the number of connected free regions that can be built out of n free monominoes (m=1). These are the "free polyominoes".

A056785(n) is the number of connected free regions that can be built out of n free dominoes (m=2). These are the "free polydominoes".

I think it would be most satisfying to submit a sequence that continues to generalize (A000105, A056785, ...). That is, either of:

- Axxxxxx(n) = 2, 28, ... is the number of connected free regions that can be built out of n (not necessarily distinct) free trominoes (m=3). These are the "free polytrominoes".

- Axxxxxx(n) = 1, 6, ... is the number of connected free regions that can be built out of n (not necessarily distinct) free straight trominoes. One might christen these the "free polybars of order m=3".

Two days ago I wrote a computer program to count "polybar" animals, and blogged the results.

https://quuxplusone.github.io/blog/2025/08/23/polybars/

The number of n-bars of order p (copied from my blog post) is:

      n=1   2    3      4        5         6         7          8         9    10
 monobars  tribars       pentabars           heptabars             nonabars
       dibars   tetrabars           hexabars             octabars        decabars

p=1:    1   1    2      5       12        35       108        369      1285  4655
  2:    1   4   23    211     2227     25824    310242    3818983  47752136
  3:    1   6   55    833    14378    269710   5221076  103352306
  4:    1   7   93   1973    47161   1204744  31711028
  5:    1   9  144   3913   118842   3851349
  6:    1  10  204   6809   252797   9951844
  7:    1  12  277  10938   478377  22178331
  8:    1  13  359  16427   830085
  9:    1  15  454  23577  1348694
 10:    1  16  558  32491  2079909

The first row of this table is OEIS A000105; the second row is A056785. None of the other rows or columns appear in the OEIS, although the second column has a simple closed form.

–Arthur