Computing derangement probabilities of the symmetric group acting on k-sets

Abstract

Let $i(\infty,k)$ be the limiting proportion, as $n \rightarrow \infty$, of permutations in the symmetric group of degree $n$ that fix a $k$-set. We give an algorithm for computing $i(\infty,k)$ and state the values of $i(\infty,k)$ for $k \le 30$. These values are consistent with a conjecture of Peter Cameron that $i(\infty,k)$ is a decreasing function of $k$.