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

Pontus von Brömssen pontus.von.bromssen at gmail.com
Sun Aug 27 14:28:17 CEST 2023

```For anyone interested, I just submitted the following sequences:
- A365139 (binary codes of polycubes)
- A365140 (binary codes of 4-d polyhypercubes)
- A365141 (binary codes of 5-d polyominoes)
- A365142 ("prime exponent codes" for polyominoes in arbitrary dimension)
- A365143 (proper dimensions of the polyominoes listed in A365142).

/Pontus

On Sun, Aug 20, 2023 at 5:16 PM Pontus von Brömssen <
pontus.von.bromssen at gmail.com> wrote:

> 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
>>
>
>
```