[seqfan] Re: simplicial polyhedra, aka maximal planar graphs: is A000109 always equal to A007030 plus A115340?

[Allan Wechsler]

Just going by the text at all three sequences, the identity looks right,
although I think different indexing choices were made for  the three
A115340. William P. Orrick has already contributed the identity at A115340.
A000109 counts all the fully-triangulated polyhedra with n vertices;
A115340 counts the number with n+2 vertices that admit a Hamiltonian cycle
(which is most of them, for small n); and A007030 is the rest of them, the
triangulated polyhedra on n+2 vertices that *don't *have a Hamiltonian
cycle. If the identity lets us fill in some values on any of the sequences,
we should do so.

All three of these sequences would be well-served by having some examples
-- imagery would be especially nice. I want to see that minimal
non-Hamiltonian deltahedron with 13 vertices.

On Sat, Feb 20, 2021 at 5:40 AM William Orrick <will.orrick at gmail.com>
wrote:

> Dear SeqFans,
>
> The subject line says it all. I suspect, but am not sufficiently conversant
> with definitions to be sure, that the relation should always hold. If
> true, A007030 can be extended. Some cross refs could also be added to all
> three sequences.
>
> -Will
>
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