[seqfan] Re: Number of connected simplicial complexes

[ALLOUCHE Jean-paul]

Well it is also true that for every pair (x,y) of distinct vertices there is no
edge between them (there are no such pairs, so it is equally true that there
is no vertex or that there is a vertex). As formulated "intuitively" by some
people "the empty set has all properties".

not quite easy to get convinced after all, but...

best
jean-paul
________________________________________
De : SeqFan [seqfan-bounces at list.seqfan.eu] de la part de Charles Greathouse [charles.greathouse at case.edu]
Envoyé : mardi 18 août 2015 15:51
À : Sequence Fanatics Discussion list
Objet : [seqfan] Re: Number of connected simplicial complexes

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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> > >
> >
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> >
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> >
>
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