[seqfan] Re: Number of connected simplicial complexes

[Charles Greathouse]

I'm curious, what are the arguments for the empty graph being disconnected?
For every pair (x, y) of distinct vertices there is an edge between them.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

On Thu, Aug 13, 2015 at 12:28 AM, Neil Sloane <njasloane at gmail.com> wrote:

> Benoit,
> Very nice! I'll update the OEIS accordingly. The two
> new entries will be A261005, A261006.
>
> (I don't feel strongly about it, but one could
> argue that the empty graph IS connected,
> as in A001349, see the comments there,
> since "the empty set has every property".
> I think the arguments for and against are about equally strong!)
>
> Best regards
> Neil
>
> Neil J. A. Sloane, President, OEIS Foundation.
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
>
>
> On Wed, Aug 12, 2015 at 3:47 PM, Benoît Jubin <benoit.jubin at gmail.com>
> wrote:
>
> > Dear Neil,
> >
> > I would say that the next term is 157. Indeed, the number of simplicial
> > complexes with n vertices is (starting at n=0):
> > 1, 1, 2, 5, 20, 180, 16143, 489996795, ...
> > It is in the OEIS... and well hidden in it. It is A006602, see comment
> > by... Vladeta Jovovic. Since the zeroth term differs, it probably
> deserves
> > its own entry.
> > Then, to count the connected ones, there is the usual summation over
> > partitions:
> > a(n) = \sum_{n_1+\dots+n_p=n, 1 \leq n_1 \leq n_2 \dots} \prod_{i=1}^p
> > a_{conn}(n_i)
> > so the connected version is given by
> > 0, 1, 1, 3, 14, 157, ...
> > and the next two terms can be easily computed.
> > Note that a_{conn}(0)=0 since the empty complex/graph/space is not
> > connected.
> >
> > Best regards,
> > Benoît
> >
> >
> > On Wed, Jul 8, 2015 at 10:35 PM, Neil Sloane <njasloane at gmail.com>
> wrote:
> >
> > > Dear Seq Fans, Back in 1983 the physicist Greg Huber
> > > sent me the initial terms of four sequences that arise in
> > > the enumeration of simplicial complexes. The second one
> > > is now A048143, and you can see an annotated
> > > and corrected scan of his letter there.
> > > But what about the first one? This is the number of
> > > unlabeled connected simplicial complexes on n nodes. It
> > > begins 1,1,3,14. It seems like a fundamental sequence in geometry.
> > > If someone could find one or two more terms, we could add it to the
> OEIS.
> > >
> > > Vladeta Jovovic, where are you when we need you?
> > >
> > > Greg also mentions two other sequences arising from his study
> > > of the early universe, but their definitions are not very explicit.
> > >
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