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

John Mason masonmilan33 at gmail.com
Sun Aug 20 09:46:38 CEST 2023

```Sorry, I see what you mean now.
I hope you choose the graded colex (87) version, as that's what I've been
using.
John

On Sun, Aug 20, 2023 at 3:33 AM Pontus von Brömssen <
pontus.von.bromssen at gmail.com> wrote:

> No, it's not independent of the ordering of the triples, basically because
> not correspond to any rotation/reflection.
>
> For the 3-dimensional V-pentomino I mentioned, the orientation that gives
> the minimum binary code in graded colex order is (the cells it occupies are
> marked with "X"):
>   0. 000 X
>   1. 100 X
>   2. 010 X
>   3. 001
>   4. 200 X
>   5. 110
>   6. 020 X
>   7. 101
>   8. 011
>   9. 002
> This means that the binary code is 2^0+2^1+2^2+2^4+2^6 = 87 in this case.
> In graded reverse lex order, it turns out that the minimum is given by the
> same orientation (I'm sure there exist other examples where this is not the
> case), but the numbers of the triples are not the same:
>   0. 000 X
>   1. 100 X
>   2. 010 X
>   3. 001
>   4. 200 X
>   5. 110
>   6. 101
>   7. 020 X
>   8. 011
>   9. 002
> So the binary code is 2^0+2^1+2^2+2^4+2^7 = 151 in this case.
>
> Cheers,
>
> Pontus
>
> On Sat, Aug 19, 2023 at 7:25 PM John Mason <masonmilan33 at gmail.com> wrote:
>
> > 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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> > >
> >
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> >
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```