[seqfan] Re: Number of connected simplicial complexes

[Benoît Jubin]

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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