Number of permutations with a single fixed point

Question

I know that the number of permutations with no fixed points over a set with $n$ elements approaches $\frac{n!}e$ as $n$ grows.

I'm interested in finding a limit (if one exists) for the number of permutations with a single fixed point.

Thank you.

Answer 1

If $D(n)$ denotes the number of derangements of $n$ (permutations with no fixed points), then a permutation with a single fixed point is just a fixed point together with a derangement of the non-fixed points, so there are $n D(n-1)$ such permutations on $n$ letters. Hence there are asymptotically $\frac{n!}{e}$ such permutations as well.

A more general approach to questions of this type that also allows you to place constraints on the number of $k$-cycles for any $k$ is given by the exponential formula; see this series of blog posts for an introduction.