Local probabilities for random permutations without long cycles

TL;DR

This paper derives asymptotic formulas for the probability that a random permutation has only small cycles, using advanced analytic methods to correct previous formulas for small cycle lengths.

Contribution

It provides new asymptotic formulas for cycle length probabilities in permutations, correcting earlier results for small cycle lengths using saddle point and number theory techniques.

Findings

Derived asymptotic formulas for cycle length probabilities

Corrected previous formulas for small r

Applied saddle point method and analytic number theory

Abstract

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.