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

TL;DR

This paper presents an algorithm to compute the limiting proportion of permutations fixing a k-set in the symmetric group, providing explicit values for k up to 30 and supporting a conjecture about their monotonicity.

Contribution

The paper introduces a novel algorithm for calculating derangement probabilities in the symmetric group and computes explicit values up to k=30, confirming a conjecture about their decreasing trend.

Findings

Computed values of i(∞,k) for k ≤ 30.

Values support Cameron's conjecture that i(∞,k) decreases with k.

Algorithm provides a practical method for derangement probability calculations.

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 $k$.