Permutations without long or short cycles

We explore probabilities that a permutation sampled from a finite symmetric group uniformly at random has only short or long cycles. Asymptotic formulas, as the order of the group increases, valid in specified regions are obtained using the saddle point method. As an application, we establish a formula with remainder term estimate for the total variation distance between the count process of the multiplicities of cycle lengths in the random permutation and a relevant process defined via independent Poisson random variables.