Permutations with Orders Coprime to a Given Integer - Bamberg, John; Glasby, S. P.; Harper, Scott; Praeger, Cheryl E. (The Electronic Journal of Combinatorics, 2020)
Let m be a positive integer and let ρ(m,n) be the proportion of permutations of the symmetric group Sym(n) whose order is coprime to m. In 2002, Pouyanne proved that ρ(n,m)n^1-φ(m)m κ_m where κ_m is a complicated (unbounded) function of m. We show that there exists a positive constant C(m) such that, for all n ≥ m, \[ C(m) (nm)^φ(m)m-1 ≤ ρ(n,m) ≤ (nm)^φ(m)m-1 \] where φ is Euler’s totient function.