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Consider a cyclically ordered collection of $r$ equinumerous agent sets withstrict preferences of every agent over the agents from the next agent set. Aweakly stable cyclic matching is a partition of the set of agents into disjointunion of $r$-long cycles, one agent from each set per cycle, such that thereare no destabilizing $r$-long cycles, i.e. cycles in which every agent strictlyprefers its successor to its successor in the matching. Assuming that thepreferences are uniformly random and independent, we show that the expectednumber 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 wherepreference list of each agent forms a partially ordered set. Each partial orderis an intersection of several, $k_i$ for side $i$, independent, uniformlyrandom, strict orders. For $k_1+k_2>2$, the expected number of stable matchingsis analyzed for three, progressively stronger, notions of stability. Theexpected number of weakly stable matchings is shown to grow super-exponentiallyfast. In contrast, for $\min(k_1,k_2)>1$, the fraction of instances with atleast one strongly stable (super-stable) matching is super-exponentially small.

On random stable matchings: cyclic matchings with strict preferences and two-side matchings with partially ordered preferences