Simplicial complices on unlabelled vertices My question is about (abstract) simplicial complices.
In particular, how many are they if I consider $n$ unlabelled vertices?
For example, if $n=4$, the two complices
$$
\{\varnothing, \{1\}, \{2\}, \{3\}, \{4\}, \{1, 2\}, \{3, 4\}\}
$$
and
$$
\{\varnothing, \{1\}, \{2\}, \{3\}, \{4\}, \{2, 3\}, \{1, 4\}\}
$$
are the same, but not
$$
\{\varnothing, \{1\}, \{2\}, \{3\}, \{4\}, \{1, 2\}, \{1, 3\}\}
$$
(since the last two sides of this one intersect in one vertex).
If $n=3$, there are 5 of them (while the Dedekind number for 3 is 20).
They are:
- dim=2
$$\{\varnothing, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{1, 3\}, \{2, 3\}, \{1, 2, 3\}\}$$
- dim=1
$$\{\varnothing, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{1, 3\}, \{2, 3\}\}$$
$$\{\varnothing, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{1, 3\}\}$$
$$\{\varnothing, \{1\}, \{2\}, \{3\}, \{1, 2\}\}$$
- dim 0
$$\{\varnothing, \{1\}, \{2\}, \{3\}\}$$
Since this last observation, I think that the answer is not the Dedekind number, but please prove me wrong if you think it is.
Thank you in advance,
Davide
PS: I am not sure whether or not this question is related to this other one. If so, please can you explain why?
PPS: I posted this question also on Math.SE, but no one answered.
 A: This is http://oeis.org/A006602 .  I checked the numbers there.
The correspondence is because the maximal elements form an antichain cover.
A: Since Brendan has identified the sequence and provided values for small $n$, let me point out that the asymptotic behavior of your sequence $s(n)$ will be $$s(n) \sim \frac{1}{n!}d(n)$$ where $d(n)$ is the $n$-th Dedekind number. 
To see why, note that a simplicial automorphism must at least induce a graph automorphism of the 1-skeleton, but the number of graph automorphisms of a graph on $n$ vertices are almost surely zero unless the number of edges is within $\log(n)$ of the extreme values $0$ and $\binom{n}{2}$. See Chapter 9 of Bollobas' book on Random graphs for a proof.

To answer your post-script -- this is very different from my question on homotopy classes of simplicial complexes in the sense that you are seeking isomorphism classes whereas I was only seeking homotopy classes. In particular, I would expect (at least for small $n$) for your sequence to be much larger than mine since there are many homotopy-equivalent but non-isomorphic simplicial complexes on $n$ vertices.
