[seqfan] Re: Number of connected simplicial complexes

[Max Alekseyev]

I'd like to comment on "the empty set has every property". This is not
quite true and, in fact, depends on the property. To figure out whether or
not the empty object (in particular, the empty set) has a particular
property, the property first needs to be formalized as a logic formula with
quantifiers. For the empty object, every universal quantifier (∀) is
evaluated to 'true' and every existential quantifier (∃) is evaluated to
'false'.

For example, we may ask if the empty permutation represents a derangement.
For permutation p being a derangement can be formalized as ∀ i∈D(p): p(i) ≠
i, where D(p) is the permutation domain of p. For the empty permutation e,
we have D(e) = empty set and thus "∀ i∈D(e): ..." is evaluated to 'true',
not matter what is hidden behind "...". So the empty permutation is
derangement.
It can be also shown that the empty permutation represents the identity (∀
i∈D(p): p(i)=i). We need to be careful here as no non-empty permutation can
be derangement and identity at the same time.

Regards,
Max

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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