[seqfan] Re: Polycubes and canonical order of k-tuples of nonnegative integers

[John Mason]

Hi
I am not sure I understand the problem.
In the case of polyominoes, taking for example the asymmetric L-shaped
tetromino, you can place it on the plane, pushed against the axes, in 8
different ways.
So you generate the 8 different representations of the tetromino and choose
the lowest one.
You can apply the same process to generating representations of polycubes,
though the number of different representations will be up to 48, according
to symmetry.
So the mechanism is independent of how you order the triples.
john

On Sat, Aug 19, 2023 at 5:17 PM Pontus von Brömssen <
pontus.von.bromssen at gmail.com> wrote:

> Hello,
>
> I just wanted to double check that the OEIS canonical way to order k-tuples
> of nonnegative integers is first by sum, then colexicographically, as
> stated in A144625 (for triples). The canonical way to order integer
> partitions (A080577) and compositions (A066099) seem to be by sum, then
> reverse lexicographic, so it seemed a little inconsistent to me.
>
> The reason I ask is that I'm preparing a 3-dimensional analog of A246521: a
> list of the binary codes of polycubes. The code depends on an ordering of
> the triples of nonnegative integers. As long as we restrict the ordering
> options to graded (i.e., first by sum) normal/reversed lex/colex, the
> lex/colex choice doesn't matter because it just corresponds to a reordering
> of the axes. In 2 dimensions, the normal/reversed option doesn't matter
> either (because reverse colex = lex), but in higher dimensions it does. For
> example the binary code of the 3-dimensional V-pentomino is 87 in normal
> (lex or colex) order, but 151 in reverse order.
>
> Of course, I could submit both normal and reversed versions, but it would
> still be nice to have one of them "canonical". It could be used to list
> properties of the polycubes, like A335573 does for 2-dimensional
> polyominoes.
>
> Best regards,
>
> Pontus von Brömssen
>
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