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

[Pontus von Brömssen]

No, it's not independent of the ordering of the triples, basically because
the permutation relating graded lex-order and graded inverse lex-order does
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
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
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