Combinatorial problems connected with P. Hall's collection process

Let $M_1, \ldots, M_r$ be nonempty subsets of any totally ordered set. Imposing some restricitons on these subsets, we find an expression for the number of elements $(\lambda_1, \ldots, \lambda_r) \in M_1 \times \cdots \times M_r$ that satisfy the condition $C$, where $C$ is a propositional formula consisting of such conditions as $\lambda_i=\lambda_j$, $\lambda_i<\lambda_j$, $i,j \in \overline{1,r}$.