# On permutations of order dividing a given integer

## TL;DR

This paper analyzes the proportion of permutations in the symmetric group whose order divides a given integer, providing detailed asymptotic results for large n and m within linear bounds.

## Contribution

It offers a detailed asymptotic analysis of permutation orders dividing a given integer in symmetric groups for large n and m.

## Key findings

- Proportion of permutations with order dividing m approaches a specific limit as n grows.
- Explicit formulas for the proportion in terms of n and m.
- Asymptotic behavior characterized for m = O(n).

## Abstract

We give a detailed analysis of the proportion of elements in the symmetric group on $n$ points whose order divides $m$, for $n$ sufficiently large and $m \ge n$ with $m = O(n)$.