[seqfan] Re: Number of connected simplicial complexes

Tue Aug 18 23:25:36 CEST 2015

Yip Maximilian is right in the sense that my argument was wrong and
that I am not an expert logician to find a better argument :-)
On the other hand the pragmatic approach of Benoit has its advantages: 
it could probably be turned into an unambiguous `a la Bourbaki definition.
[The case of p prime if and only if it is divisible only by itself and by 1,
is easily "slightly" transformed to eliminate 1 by saying "if and only if
it has exactly two divisors".]

best wishes
jean-paul
________________________________________
De : SeqFan [seqfan-bounces at list.seqfan.eu] de la part de Benoît Jubin [benoit.jubin at gmail.com]
Envoyé : mardi 18 août 2015 23:19
À : Sequence Fanatics Discussion list
Objet : [seqfan] Re: Number of connected simplicial complexes

The question is rather: "Is it more convenient to define the empty graph as
connected or not?" (in the sense that more theorems are then stated without
the need to deal with special cases).

As Frank Adams-Watter wrote, this is similar to 1 being prime or not:
unique decomposition as a product of primes / as a disjoint union of
connected components. Also, the first comment in http://oeis.org/A001349,
"but with a(0) omitted" acknowledges that the empty graph should not be
counted as connected: this would simplify the statements of mathematical
results.

In this respect, I recommend the reading of
http://math.stackexchange.com/questions/50551/is-the-empty-graph-connected
and
http://ncatlab.org/nlab/show/too+simple+to+be+simple
and
http://mathoverflow.net/questions/120536/is-the-empty-graph-a-tree

Regards,
Benoît

On Tue, Aug 18, 2015 at 10:11 PM, M. F. Hasler <oeis at hasler.fr> wrote:

> But the definition of disconnected is not "all vertices..." but "there
> exist..."
> e.g.: from mathworld:
> "A graph is said to be disconnected if it is not connected, i.e., if there
> exist two nodes such that no path..."
>
> and for the empty graph there are no such two nodes...
>
> This is a little bit different from the situation of e.g.the empty set
> which is open and closed at the same time.
>
> So I would have answered like Charles and I would be astonished if a
> "serious" reference work would consider the empty graph as not connected...
>
> Maximilian
> Le 18 août 2015 21:41, "ALLOUCHE Jean-paul" <Jean-paul.ALLOUCHE at imj-prg.fr
> >
> a écrit :
>
> > 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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