On recurrence relation in the problem of enumeration of finite posets

In the previous paper of the author the formula reduced the count of the number $T_0(n)$ of posets defined on $n$-set to the calculation of the numbers $W(p_1,\ldots,p_k)$ of posets of a special form has been proved ($p_1+\ldots+p_k=n$). In present paper we obtain the relations of recurrent nature connecting the individual values of $W(p_1,\ldots,p_k)$ among themselves. As a result of these relations the partially folded formula for the number $T_0(n)$ is obtained.