[seqfan] Re: Bad pdf file in A002369?

Fred Lunnon fred.lunnon at gmail.com
Sun Feb 25 01:18:09 CET 2018


  Incidentally, when enumerating objects acted upon by a symmetry group
--- eg. polyominoes under the group  D_4 (aka  D_8 ) of the square --- it is
advisable to consider "fixed" counts (ignoring symmetry) rather than "free"
(modulo symmetry), since any explicit expressions for the counts are
generally simpler for the fixed case.

  In practice, brute-force enumeration may well be faster if symmetry
is `broken' first; but it is generally easy to incorporate additional code
to reconstruct the fixed contribution corresponding to each free object.

  Unfortunately many investigators remain apparently unaware of such
considerations.

WFL



On 2/24/18, Joseph Myers <jsm at polyomino.org.uk> wrote:
> On Sat, 24 Feb 2018, Richard J. Mathar wrote:
>
>> The construction of sequence A002369 is not clear to me either
>> (probably because I don't understand the fine-tuned French).
>> A056780 counts the shapes of polyominoes of rectangular cells, where
>> polyominoes are considered equivalent if they can be
>> mapped onto each other by the symmetry group of the
>> rectangle of order 4 (unity, 180 degrees rotations, horizontal flip,
>> vertical flip).
>> Devisme seems to say he has counted A002369 by counting a tree-structure
>> and by considering paths stepping across connected cells horizontally
>> or vertically.
>> So which shapes of A056780 are not counted in A002369 and why?
>
> If it's specifically meant to be strips of stamps, one might suppose that
> some internal edges of the polyomino are to be severed so that the graph
> of the result is a path.  This would result in two variants of the square
> tetromino instead of one, but both variants of the T tetromino would be
> removed, so resulting in the count of 8.  Unfortunately, that (which if
> you start with squares rather than rectangles would be A037245 with an
> additional leading term of 1) doesn't agree with all the other figures.
> I get: 1, 2, 3, 8, 16, 44, 106, 294, 762, 2094, 5572, 15200, 40778,
> 110626, 297236, 803258, 2156828, 5811502.
>
> --
> Joseph S. Myers
> jsm at polyomino.org.uk
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>


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