%0 Journal Article
%T Conditional probability of derangements and fixed points
%J Transactions on Combinatorics
%I University of Isfahan
%Z 2251-8657
%A Gutmann, Sam
%A Mixer, Mark D.
%A Morrow, Steven
%D 2023
%\ 03/01/2023
%V 12
%N 1
%P 11-26
%! Conditional probability of derangements and fixed points
%K derangement
%K Fixed Point
%K probability
%R 10.22108/toc.2022.131705.1941
%X 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.
%U https://toc.ui.ac.ir/article_26291_f825454f21ba5f39cbf58e4058b7d906.pdf