[seqfan] Re: Polyominoes (A000105, A000988, A001168) - offset problem

Oscar Cunningham mail at oscarcunningham.com
Thu Jun 25 15:38:36 CEST 2020


Hi all,

One of the defining features of polyominos is that they're connected. 
But I believe the correct convention is that the empty set is not 
connected. This is for the same reason that 1 is not a prime number. 
Good expositions of this idea are found in Harary and Read's 'Is the 
null-graph a pointless concept?' 
(https://link.springer.com/chapter/10.1007/BFb0066433) and the nLab page 
'too simple to be simple' 
(https://ncatlab.org/nlab/show/too+simple+to+be+simple).

So we should say that there is no polyomino of size 0.

Best,

Oscar Cunningham

On 25/06/2020 13:15, Jean-Luc Manguin wrote:
> Hello everyone,
>
> Thank you Brendan to "bring some water to my mill", and I am sorry to "shake the tree up to the top", but a request (actually 3) on OEIS with "number of fixed/one-sided/free polyominoes" give results that show this question is not definitely clear.
> Moreover, the sequences such as A292357 could be strangely impacted if we admit that there is a "zeromino" ; in that case, we should complete the first diagonal with a "1" at the beginning (if I follow your hypothesis), but what should we put in the corresponding line and row ?? And what could mean "a polyomino with height = 0 and width = n" ??
> And moreover again, why should we consider there is ONE zeromino ? Why not zero ?
>
> The usual "rule" is (IMO) to exhibit a publication which justify such a choice ; till now, I have seen nothing.
>
> Thank you for your attention.
> Best regards,
>
> JLM
>
> ----- Mail original -----
> De: "Brendan McKay" <Brendan.McKay at anu.edu.au>
> À: "seqfan" <seqfan at list.seqfan.eu>
> Envoyé: Jeudi 25 Juin 2020 11:15:55
> Objet: [seqfan] Re: Polyominoes (A000105, A000988, A001168) - offset problem
>
> Hi Neil,
>
> Anyone who sees an error on Wikipedia, and knows how to fix it but
> doesn't, is
> responsible for the error.
>
> But in this case there is no error.  Wikipedia follows "reliable
> sources" and in this case
> the reliable source is Golumb's book "Polyominoes".  From a brief look
> it seems that
> Golumb does not recognise a null polyomino (see Fig 1 and Tables 3 and
> D.1), so
> Wikipedia is *required* to either also disallow it or to change to a
> different source.
>
> Incidentally, the first comment on A0000105, which is
> "A0000105(n) + A030228(n) = A000988(n)", is not correct for n=0 at the
> moment.
>
> Cheers, Brendan.
>
> On 25/6/20 5:27 pm, Neil Sloane wrote:
>> Everything in the Wikipedia is inaccurate.
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>
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