Recent results on permutations without short cycles

TL;DR

This paper investigates the density of permutations in symmetric groups that lack short cycles, providing new asymptotic formulas for this density using advanced mathematical techniques.

Contribution

It introduces novel asymptotic formulas for the density of permutations without short cycles, expanding understanding in permutation cycle structure analysis.

Findings

Derived new asymptotic formulas for permutation densities

Applied saddle-point method to analyze cycle restrictions

Enhanced mathematical understanding of permutation cycle distributions

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 $5\leq r< n$ and $n\to\infty$.