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

[D. S. McNeil]

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://www.combinatorics.org/ojs/index.php/eljc/article/download/109/pdf
>
> https://www.researchgate.net/profile/Marco-Pellegrini-15/publication/337969707_Further_progress_on_the_Buratti-Horak-Rosa_conjecture/links/5dfb143492851c836488482a/Further-progress-on-the-Buratti-Horak-Rosa-conjecture.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.
>
>