The number of cycles in random permutations without long cycles is   asymptotically Gaussian
  For uniform random permutations conditioned to have no long cycles, we prove that the total number of cycles satisfies a central limit theorem. Under additional assumptions on the asymptotic behavior of the set of allowed cycle lengths, we derive asymptotic expansions for the corresponding expected value and variance. 
Volker Betz and Helge Sch\"afer
math.PR