  We explore the probability that a permutation sampled from the symmetric
group of order n uniformly at random has cycles of lengths not exceeding r.
Asymptotic formulas valid in specified regions for the ratio n/r are obtained
using the saddle point method combined with ideas originated in analytic number
theory. Theorem 1 and its detailed proof are included to rectify formulas for
small r which have been announced by a few other authors.
