Indecomposable permutations with a given number of cycles

A permutation a1a2 . . . an is indecomposable if there does not exist p < n such that a1a2 . . . ap is a permutation of {1, 2, . . . , p}. We compute the asymptotic probability that a permutation of Sn with m cycles is indecomposable as n goes to infinity with m/n fixed. The error term is O( log(n−m) n−m ). The asymptotic probability is monotone in m/n, and there is no threshold phenomenon: it degrades gracefully from 1 to 0. When n = 2m, a slight majority (51.1 . . . percent) of the permutations are indecomposable. We also consider indecomposable fixed point free involutions which are in bijection with maps of arbitrary genus on orientable surfaces, for these involutions with m left-to-right maxima we obtain a lower bound for the probability of being indecomposable.