[seqfan] Re: Maximally symmetric "even" polyominoes

[Allan Wechsler]

Thank you, John Mason, and indirectly, Robert A. Russell.

John, your formula for the proposed A(n) has to be right, because it is
counting the ways a sufficiently symmetric polyomino can be slotted into
the upper right quadrant, so that when replicated into the other three
quadrants it forms a maximally-symmetric polyomino. One with single
diagonal symmetry can be slotted in in two ways, which differ by reflection
around the other diagonal axis, the one about which the given polyomino is
not symmetric. One with double diagonal symmetry (but no orthogonal
symmetry) can also be placed in two ways, which differ by a 90-degree
rotation. Finally, maximally symmetric polyominoes can be placed in a
single way. This gives the coefficients you noted, 2P + 2Q + R. (Also,
you're quite right that I missed one for n=5.)

I won't have time to construct an entry until this weekend at the earliest.
John, feel free to jump in first. I think you should get authorship credit,
given your earlier work.

(Isn't it striking how the even entries lag the odd ones? This is because
an even-order diagonally-symmetric polyomino must have an even number of
cells on the symmetry axis -- that is, it has to have at least 2, while an
odd-order example can have 1. This spares more cells in the odd-order case
for messing around off-axis.)

On Thu, Jan 27, 2022 at 4:30 AM John Mason <masonmilan33 at gmail.com> wrote:

> On second thoughts, Robert A. Russell can certainly help, as he authored
> sequence A346800, which has a completely different definition but which
> apparently (needs proof) corresponds to your desired sequence.
> (See values below)
> Last year I formulated but didn’t prove the theory that (your proposed
> sequence) a(n) = 2*A006748(n) + 2*A056878(n) + A142886(n).
> Where A006748 enumerates singly 45 degree reflectively symmetric
> polyominoes and A056878 enumerates those doubly so.
> That said, the values you are looking for should be those below.
> As you see, I have a different value for a(5) corresponding to size 20.
> I find these (use Courier New):
>
> __OO__
> __OO__
> OOOOOO
> OOOOOO
> __OO__
> __OO__
>
> OOOOOO
> O____O
> O____O
> O____O
> O____O
> OOOOOO
>
> _OOOO_
> OO__OO
> O____O
> O____O
> OO__OO
> _OOOO_
>
> __OO__
> _OOOO_
> OO__OO
> OO__OO
> _OOOO_
> __OO__
>
> _O__O_
> OOOOOO
> _O__O_
> _O__O_
> OOOOOO
> _O__O_
>
> That said, I think that it should be sufficient to check the above, and
> then introduce a comment in Robert’s sequence to say that it corresponds
> ALSO to what you were looking for.
>
> john
>
>
>
> 1
> 0
> 2
> 1
> 5
> 4
> 16
> 13
> 54
> 46
> 186
> 167
> 660
> 612
> 2384
> 2267
> 8726
> 8464
> 32278
> 31822
> 120419
> 120338
> 452420
> 457320
> 1709845
> 1745438
> 6494848
> 6686929
> 24779026
> 25703792
> 94899470
> 99096382
> 364680344
> 383067646
> 1405619344
> 1484352159
> 5432421429
> 5764277096
> 21046198560
> 22429257682
> 81716371069
> 87432657722
> 317917129256
> 341394729018
> 1239120776640
> 1335080732960
> 4837744188806
> 5228480834780
>
>
>
> 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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