[seqfan] Re: Polyomino miscellany

[Alex Meiburg]

Hmm, I'd agree, on the basis that zero-choose-one is either zero or
undefined.

On Wed, Dec 26, 2018, 11:03 PM Allan Wechsler <acwacw at gmail.com wrote:

> Earlier I posted about my misadventures trying to write a Haskell program
> to compute A048664, which counts oriented polyominoes with a particular
> cell chosen as the "origin". Rotations, reflections, and different choices
> of the origin cell are all considered different.
>
> I had a bug in my program, which produced a sequence that grew quite a bit
> faster than A048664 -- the first discrepancy was that a(3) was 24 instead
> of 18.
>
> I found the bug, and now know what I was counting, and the concept seems
> interesting enough to add to OEIS, so I will probably do that in a few
> days. For the moment I'm going to keep my explanation secret, in case
> anyone wants to have fun figuring out the "secret rule" (knowing that it
> arose from a buggy polyomino-counter).
>
> (I fixed the bug, as well, and the program now produces A048664 just fine
> -- except that it in excruciatingly slow.)
>
> While fossicking among the polyomino sequences, I noticed A126202, which
> also counts polyominoes with a single distinguished cell, but this sequence
> regards rotations and reflections as equivalences. I was intrigued to note
> that, unlike the extremely similar A048664, A126202 has an offset of 0; it
> says there is one pointed nullomino.
>
> Now, I am an enthusiastic supporter of the Zero Liberation Front. Zero is a
> good number, the empty set is a a good set, and the nullomino is a good
> polyomino, and should be counted wherever this can be remotely justified.
> But I cannot see the justification for counting it (more than zero times)
> in the case of A126202. The problem is that it has no cells from among
> which to select the "origin". There is a nullomino, yes -- but there is no
> nullomino with exactly one distinguished cell. In this case, I think
> A048664 (which is 1-offset) got it right. If a(0) must be defined in either
> case, I think, the only defensible value is 0, because there are no objects
> satisfying the definition. Can anyone defend A126202(0) = 1?
>
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