Region distributions of graph embeddings and stirling numbers

AbstractIt is shown that the distribution of the number of regions r in the random orientable embedding of the graph with one vertex and q loops is approximately proportional to the unsigned Stirling numbers of the first kind s(2q,r) where r has different parity from q. This approximation is strong enough to imply that both the limiting mean and variance of this distribution differ from ln 2q by small known constants. The paper concludes with a result on the unimodality of some recursively defined sequences and also some conjectures regarding region distributions of arbitrary graphs