# [seqfan] Re: Maximally symmetric "even" polyominoes

John Mason masonmilan33 at gmail.com
Thu Jan 27 09:34:09 CET 2022

I bet Robert A. Russell can help you.
john

From: Allan Wechsler
Sent: 27 January 2022 03:32
To: Sequence Fanatics Discussion list
Subject: [seqfan] Maximally symmetric "even" polyominoes

Some polyominoes have all the symmetry it is possible to have on a square
grid. The number of such maximally-symmetric polyominoes is counted by
https://oeis.org/A142886.

These polyominoes can be divided into two classes. First are the ones
typified by the monomino, which are centered on a cell. Then there are
those typified by the square tetromino, which aren't. I want to call these
"odd" and "even", respectively, but I fear this could be confusing, because
although the "even" class all have an even number of cells, the "odd" class
can have either an odd or an even number of cells. So maybe "face-centered"
and "vertex-centered"? Advice is welcome.

The "vertex-centered" examples all have order that is a multiple of 4. With
4 cells, the square tetromino is the lone example. With 8 cells, there are
no examples. With 12 cells, there are two different maximally-symmetric
dodecominoes. (For those who know how to decode these, they have encodings
F99F and 6FF6. Details on request.)

With 16 cells, there is just one example.

With 20 cells, there are 4; there are 2 with 24 and, if I haven't made a
stupid mistake, there are 10 with 28 cells.

You have probably guessed that I wouldn't be blathering on about this if
1,0,2,1,4,2,10... was in OEIS. You're right. It isn't. I think it should
be, but my polyomino-counting ability is poor. Does anybody have software
within easy striking distance of this census?

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