[seqfan] Re: Counting polyominoes with peculiar symmetry

[Allan Wechsler]

Yes, Alex, sorry, I wasn't clear. The T-tetromino qualifies; the two sets
of diagonals are congruent, though they are mirror images of each other.

I think these guys are of two kinds: the ones with a mirror symmetry axis
parallel to the grid running down the middle of a row of cells, and the
ones with a 180-degree rotation symmetry whose center is in the middle of
an edge. I may have missed a possibility though.

On Wed, Sep 22, 2021 at 9:25 PM Alex Meiburg <timeroot.alex at gmail.com>
wrote:

> One relevant question, when you say "isomorphic", do you only consider that
> under plane rotations? Or reflections too?
>
> For example, for a T-shaped polyomino, the two are only isomorphic if you
> include reflections.
>
> On Wed, Sep 22, 2021, 18:20 Allan Wechsler <acwacw at gmail.com> wrote:
>
> > Take a polyomino and mark each of its square cells by drawing one of its
> > two diagonals. But once you choose the first diagonal, all the others are
> > forced because I require these diagonal markings to meet at their
> > endpoints. (That is, /\ is fine, but // isn't allowed.)
> >
> > Depending on your choice of the first diagonal, you can mark up any
> > polyomino in two ways, but sometimes these two ways turn out to be
> > isomorphic. That only happens for polyominoes that have some symmetry,
> but
> > it doesn't happen for all symmetric polyominoes, just some of them.
> >
> > By my hand count, the number of qualifying polyominoes of orders 1
> through
> > 6 are:
> >
> > 1,1,1,3,4,10
> >
> > But I am not sure of the hexomino number. None of the 11 matches at OEIS
> > say anything about polyominoes.
> >
> > I hope some polyomino counters out there understand what I mean, and can
> > verify or contradict my counts, and tell me whether the resulting
> sequence
> > is already in OEIS or not.
> >
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> >
>
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