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

Pontus von Brömssen pontus.von.bromssen at gmail.com
Sun Aug 20 17:16:47 CEST 2023

```Just a small clarification of the code I proposed for polyominoes of
arbitrary dimension: the mapping should be (x_1, x_2, ...) ->
(Product_{i>=1} prime(i)^x_i) - 1. The "-1" in order to get the exponents
of 2 to start at 0.

By the way, maybe this code has been used before; does anyone (John?) know?

/Pontus

On Sun, Aug 20, 2023 at 1:21 PM Pontus von Brömssen <
pontus.von.bromssen at gmail.com> wrote:

> That's great, John! If there's a choice already in use I'll definitely use
> that. On top of that, it's the same ordering as OEIS recommends (A144625).
> (Like I wrote, the choice between lex/colex doesn't matter for the binary
> code, but I'll choose the nonreversed version.)
>
> One more question: I also plan to submit a similar sequence for
> polyominoes with no restriction on the dimension (those counted by
> A005519). To define a code similar to the binary code under discussion
> here, we need an ordering of sequences of nonnegative integers with a
> finite number of nonzero terms. There's an obvious choice for that, namely
> to interpret the sequences as prime exponents, mapping (x_1, x_2, ...) to
> the product of prime(i)^x_i for i >= 1. Then we get a binary code for
> polyominoes in any dimension by choosing the orientation that minimizes the
> sum of the powers of 2 with exponents corresponding to the labels of the
> cells of the polyomino. What would be a good name for this code? Prime
> binary code? Binary prime code? Binary prime exponent code? Any better
> suggestions?
>
> Thanks also to Marc LeBrun for good advice and kind words.
>
> All the best,
>
> Pontus
>
```