Conditional Probability of Derangements and Fixed Points

TL;DR

This paper investigates the conditional probability of fixed points in permutations, showing it decreases with parameters and can be approximated by a specific formula, extending understanding of permutation fixed point distributions.

Contribution

It introduces new results on the behavior and approximation of conditional fixed point probabilities in permutations, generalizing previous derangement probability results.

Findings

Conditional probability decreases with k and n.

Approximation formula for the probability is provided.

Results extend to fixed points given a fixed number d.

Abstract

The probability that a random permutation in $S_n$ is a derangement is well known to be $\displaystyle\sum\limits_{j=0}^n (-1)^j \frac{1}{j!}$. In this paper, we consider the conditional probability that the $(k+1)^{st}$ point is fixed, given there are no fixed points in the first $k$ points. We prove that when $n \neq 3$ and $k \neq 1$, this probability is a decreasing function of both $k$ and $n$. Furthermore, it is proved that this conditional probability is well approximated by $\frac{1}{n} - \frac{k}{n^2(n-1)}$. Similar results are also obtained about the more general conditional probability that the $(k+1)^{st}$ point is fixed, given that there are exactly $d$ fixed points in the first $k$ points.