Recent results on permutations without short cycles

Abstract

The density, denoted by $\kappa (n,r)$, of permutations having no cycles of length less than $r+1$ in a symmetric group ${\mathrm{S}}_n$ is explored. New asymptotic formulas for $\kappa (n,r)$ are obtained using the saddle-point method when $5\le r<n$ and $n\to \infty$.