On the frequency of permutations containing a long cycle

A general explicit upper bound is obtained for the proportion $P(n,m)$ of elements of order dividing $m$, where $n-1 \le m \le cn$ for some constant $c$, in the finite symmetric group $S_n$. This is used to find lower bounds for the conditional probabilities that an element of $S_n$ or $A_n$ contains an $r$-cycle, given that it satisfies an equation of the form $x^{rs}=1$ where $s\leq3$. For example, the conditional probability that an element $x$ is an $n$-cycle, given that $x^n=1$, is always greater than 2/7, and is greater than 1/2 if $n$ does not divide 24. Our results improve estimates of these conditional probabilities in earlier work of the authors with Beals, Leedham-Green and Seress, and have applications for analysing black-box recognition algorithms for the finite symmetric and alternating groups.