[seqfan] Re: Counting polyominoes with peculiar symmetry

[jean-paul allouche]

Hi

A (possibly stupid) question: is there any chance that, up to
its first terms, the sequence given just below, namely
(...) 2  4  6  12  20 be the interspersion of A000984 = (..) 2  6 20...
and twice A000984 = (...) 4  12  40...?

best
jp


Le 23/09/2021 à 05:13, Allan Wechsler a écrit :
> 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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