Geodesic
On enumeration of posets defined on finite set
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 13 (2016), pp. 318-330.

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If $T_0(n)$ is the number of partial orders (labeled $T_0$-topologies) defined on a finite set of $n$ elements then the formula hold $$ T_0(n)=\sum\limits_{p_1+\ldots+p_k=n} (-1)^{n-k}\,\frac{n!}{p_1!\ldots p_k!}\,W(p_1,\ldots,p_k), $$ where the summation is over all ordered sets $(p_1,\ldots,p_k)$ of positive integers such that $p_1+\ldots+p_k=n$. The number $W(p_1,\ldots,p_k)$ is the number of partial orders of a special form. If $D_k$ is the dihedral group of order $2k$ then $W(p_{\pi(1)},\ldots,p_{\pi(k)})=W(p_1,\ldots,p_k)$ for all $\pi\in D_k$. We studied the complemented partial orders.
Keywords: graph enumeration, poset, finite topology. 
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V. I. Rodionov. On enumeration of posets defined on finite set. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 13 (2016), pp. 318-330. http://geodesic.mathdoc.fr/item/SEMR_2016_13_a58/

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