Local Probabilities for Random Permutations Without Long Cycles

Abstract

We explore the probability $\nu(n,r)$ that a permutation sampled from the symmetric group of order $n!$ uniformly at random has no cycles of length exceeding $r$, where  $1\leq r\leq n$ and $n\to\infty$. 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.