[seqfan] Re: Sequence proposal by John Mason (by way of moderator)

[M. F. Hasler]

Dear SeqFans,

Since that sequence seems still not yet there, I made a start with
https://oeis.org/draft/A246521
List of free polyominoes in binary coding, ordered by number of bits,
then value of the binary code. Can be read as irregular table with row
lengths A000105.

It would be easy to write a short & simple program which could
generate the n-ominos by "growing" (n-1)-ominoes, represented as sets
of the coordinates (x,y) of the squares, into all possible directions,
and making a check for congruence modulo rotation or axial symmetry.
(Imposing e.g. that the maximal y-coordinate is never larger than the
maximal x-coordinate could avoid producing/considering irrelevant
"duplicate" configurations.)

I will also add a the versions corresponding to the fixed polyomials
counted in A000988 and A001168.

Once again, more general 2-dimensional patterns could be coded in the same way,
e.g. the burst patterns counted in A093424.

Reciprocally, nonnegative integers could be classified according to what
what would be the "canonical" representation of the pattern
represented by their bitmap
(with or without "modulo" chirality and/or rotations)
and in particular one could make lists of those representing (any)
polyominoes, 1-, 2-, 3-,.. n-ominoes,
etc.
Any contributions are welcome.

Maximilian

On Mon, Aug 25, 2014 at 12:19 AM, M. F. Hasler <oeis at hasler.fr> wrote:
> I like very much Frank's prescription for coding 2-dimensional shapes,
> here polyominoes,
> by adding those powers of 2, filling the quarter-plane in the
> "(OEIS-)canonical" way (antidiagonals), which are "covered" by the
> shape (in the way which yields the smallest result).
> For the polyominoes this would yield
> 1 (.) ; 3 (..) ; 7 (:.), 11(...) ; 15 (:..), 23 (::), 27 (.:.), 30
> (.:*), 75 (....) ; &c
> The sequence
> 1,3,7,11,15,23,27,30,75, ...
> is not yet in OEIS, I can submit this (with due credits to Frank)
> unless s/o already started the work.
>
> (Thereafter, the sequence will no more be increasing,
> if one follows the prescription to list triominos, 4-ominos, 5-ominos,...:
> The first element in each series (or row, if considered as table)
> would be 2^n-1, and the last element
> woud be 2^0+2^1+2^3+2^6+...+2^T(n).)
> OTOH one could also define and list these "polyomino numbers" which
> are those numbers which represent some polyomino (using the above
> prescription!(*)),
> i.e., the former sequence re-ordered by the size of the terms.)
> (*) there would also be another [actually 2 other] sequence[s]
> (supersequence[s] of the former), of numbers which represent
> polyominos without the restriction:
> (a) of removing equivalent polyominos (e.g. there would be " .. "  AND " : ")
> (b) of translating them as to touch the x and y axis
> (i.e., e.g., the ".." and " : " could lie anywhere in the quarter-plane).
>
> PS: another way of assigning the weigths to the grid points would be
> to number them not following antidiagonals, but "filling squares":
> 0 1 4
> 3 2 5
> 8 7 6
> etc.
> This would lead to a different variants for each of the 4 above sequences.
>
> Maximilian
>
>
> On Sun, Aug 24, 2014 at 1:43 PM, Frank Adams-Watters
> <franktaw at netscape.net> wrote:
>> Sorry, I should have said the domino is represented by 3, not 2.
>>
>>
>> Franklin T. Adams-Watters
>>
>> -----Original Message-----
>> From: Frank Adams-Watters <franktaw at netscape.net>
>> To: seqfan <seqfan at list.seqfan.eu>
>> Sent: Sun, Aug 24, 2014 12:40 pm
>> Subject: [seqfan] Re: Sequence proposal by John Mason (by way of moderator)
>>
>> ... So the monomio would
>> be represented by 1, the domino by 2, and the trionimos by 7 and 11. ...
>>
>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/
>
>
>
> --
> Maximilian



-- 
Maximilian