On recursion relations in the problem of enumeration of posets

In two previous works of the author, published in this journal, a series of formulas are obtained related to the themes of enumeration of partial orders (finite topologies). In the first work, a formula is proved that reduces the calculation of the number $T_0(n)$ of all partial orders defined on an $n$-set to the calculation of the numbers $W(p_1,\ldots,p_k)$ of partial orders of a special form. In the second paper, a partially convolute formula is obtained for the number $T_0(n)$. Relations of a recurrent nature are obtained that relate the individual values $W(p_1,\ldots,p_k).$ Explicit formulas are presented for calculating the individual values $W(p_1,\ldots,p_k). $ In this paper, we obtain new recurrence relations that relate the separate numbers $W(p_1,\ldots,p_k)$ between themselves. The obtained equations are enough to calculate without the computer the numbers $T_0(n)$ for all $n<9.$ To calculate the number $T_0(9)$ of these relations not enough (the number of required numbers $W(p_1,\ldots,p_k)$ is $128$, and the rank of the system matrix is $123$; there are not enough five equations generating the desired rank). We admit the presence of some general regularity generating new formulas.