[seqfan] Re: Counting polyominoes with peculiar symmetry

[Allan Wechsler]

Given that this class of polyominoes is the union of two smaller
well-defined classes, it would make sense to see if OEIS has censuses of
each of those classes, and of their intersection.

For polyominoes with a mirror symmetry axis along an orthogonal line of
cells, I get 1,1,1,2,4,6,12,20, which isn't archived either. I counted
these by eye by looking at diagrams in the Wikipedia articles. The hard
part is excluding the orthogonal mirror-symmetric ones whose mirror axes go
*between* lines of cells. Counting these would be interesting too.

The ones with a rotational symmetry around the center of a cell edge all
have an even number of cells, so the odd entries in the sequence would be
0. For polyominoes with 2n cells and this kind of symmetry, I get 1,2,6,19,
which is not enough terms to find out anything for sure, and I don't have
an easy way to winnow through the deconimoes. Somebody with some polyomino
software will need to step in soon.

It feels like there are quite a few unmined possibilities here,
distinguishing symmetries based on where the axes and centers are with
respect to the grid.

On Wed, Sep 22, 2021 at 10:10 PM Jon Wild <wild at music.mcgill.ca> wrote:

>
> I agree with Allan's count. The rule seems to be: a polyomino qualifies,
> in Allan's sense, if it meets at least one of these two conditions:
>
> 1) it has a mirror symmetry whose axis runs through the centre of a column
> or row of cells (i.e. not along a line that forms the cell edges)
>
> 2) it has a rotational symmetry about a fixed point that is at the
> midpoint of a cell edge (i.e. not at a cell corner, and not in the middle
> of a cell)
>
> It looks to me as if there are 12 heptominoes that qualify. The sequence
> 1,1,1,3,4,10,12 is not in the encyclopedia.
>
> jon wild
>
>
> On Wed, 22 Sep 2021, Allan Wechsler 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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