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

[Pontus von Brömssen]

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