Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 there's exist) for the number of permutations with a single fixed point.

Thank you

share|cite|improve this question
I think you mean "the number of permutations over a set with $n$ elements with no fixed points approaches $\frac{n!}{e}$." – Qiaochu Yuan Jan 21 '12 at 18:57
corrected it... thanks. – Amihai Zivan Jan 21 '12 at 19:04
up vote 7 down vote accepted

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.

share|cite|improve this answer
Why D(n) is asymptotically n!/e? - Thanks – Amihai Zivan Jan 21 '12 at 19:18
@Amihai: see for example . – Qiaochu Yuan Jan 21 '12 at 19:22
Sorry, was not thinking. it's easy... thanks again! – Amihai Zivan Jan 21 '12 at 19:28

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.