# [seqfan] Re: Fwd: Worry about old sequence, A030077, paths in K_n, and new sequence A352568

Neil Sloane njasloane at gmail.com
Sun Apr 3 16:35:39 CEST 2022

```Doug, Could you log in and update A352568 accordingly? Thanks!
Best regards
Neil

Neil J. A. Sloane, Chairman, OEIS Foundation.
Also Visiting Scientist, Math. Dept., Rutgers University,
Email: njasloane at gmail.com

On Sun, Apr 3, 2022 at 10:28 AM D. S. McNeil <dsm054 at gmail.com> wrote:

>
> If I'm reading that literature correctly, then can't we extend the new
> A352568 by a few terms, because it's known that every admissible multiset
> is realizable for at least v <= 19 (in the terminology of
> https://arxiv.org/pdf/2105.00980.pdf)?
>
> I think that'd give us up to
>
> [1, 1, 1, 3, 5, 17, 28, 105, 161, 670, 1001, 4129, 6188, 26565, 38591,
> 167898, 245157, 1072730, 1562275]
>
> if we simply loop over the candidate multisets and check admissibility.
>
>
> Doug
>
> On Sun, Apr 3, 2022 at 5:09 AM Brendan McKay via SeqFan <
> seqfan at list.seqfan.eu> wrote:
>
>> Thanks to a tip from Darryn Bryant, I have found that there is
>> substantial literature
>> related to A352568, that is
>> "Take n equally spaced points on circle, connect them by a path with n-1
>> line segments;
>> sequence gives number of distinct multisets of segment [not path, Neil]
>> lengths".
>>
>> The question is: Which multisets of segment lengths can be realised?
>>
>> Baratti conjectured that all multisets can be realised if n is prime.
>> This is proved
>> up to n=23 but remains open in general.
>>
>> Baratti, Horak and Rosa conjectured a complete answer to the the
>> question and many
>> special cases have been proved.  Searching for these names gives many
>> hits, for
>> example:
>>
>> https://doi.org/10.1016/j.disc.2021.112486
>> https://arxiv.org/pdf/1311.2785.pdf
>>
>> https://doi.org/10.1016/j.disc.2013.11.017
>> https://arxiv.org/abs/2202.07733
>> http://bica.the-ica.org/Volumes/94/Reprints/BICA2021-21-Reprint.pdf
>>
>> I stopped there but there are plenty more.
>>
>> Cheers, Brendan.
>>
>>

```