  Consider a cyclically ordered collection of $r$ equinumerous agent sets with
strict preferences of every agent over the agents from the next agent set. A
weakly stable cyclic matching is a partition of the set of agents into disjoint
union of $r$-long cycles, one agent from each set per cycle, such that there
are no destabilizing $r$-long cycles, i.e. cycles in which every agent strictly
prefers its successor to its successor in the matching. Assuming that the
preferences are uniformly random and independent, we show that the expected
number of stable matchings grows with $n$ (cardinality of each agent set) as
$(n\log n)^{r-1}$. We also consider a bipartite stable matching problem where
preference list of each agent forms a partially ordered set. Each partial order
is an intersection of several, $k_i$ for side $i$, independent, uniformly
random, strict orders. For $k_1+k_2>2$, the expected number of stable matchings
is analyzed for three, progressively stronger, notions of stability. The
expected number of weakly stable matchings is shown to grow super-exponentially
fast. In contrast, for $\min(k_1,k_2)>1$, the fraction of instances with at
least one strongly stable (super-stable) matching is super-exponentially small.
