[seqfan] Re: Number of connected simplicial complexes

[Frank Adams-Watters]

The way I look at it, it's not so much that it's disconnected, as that connected is not quite the right concept. Instead, one is usually interested in single-component graphs - there exists a node from which you can trace a path to every other node, rather than for all nodes you can trace a path to every other node.

The reason for preferring this is similar to why 1 is not considered a prime: every graph has a unique decomposition into single-component graphs.

Franklin T. Adams-Watters

-----Original Message-----
From: Charles Greathouse <charles.greathouse at case.edu>
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Sent: Tue, Aug 18, 2015 8:52 am
Subject: [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.
> > >
> > >
_______________________________________________
> > >
> > > Seqfan Mailing list
- http://list.seqfan.eu/
> > >
> >
> >
_______________________________________________
> >
> > Seqfan Mailing list -
http://list.seqfan.eu/
> >
>
>
_______________________________________________
>
> Seqfan Mailing list -
http://list.seqfan.eu/
>

_______________________________________________

Seqfan
Mailing list - http://list.seqfan.eu/