On enumeration of posets defined on finite set

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.