Fixed points of random permutations

The following two results are straightforward and reasonably well-known exercises in combinatorics:

The number of permutations on Image may be NSFW.
Clik here to view. elements with no fixed points (derangements) is approximately Image may be NSFW.
Clik here to view. $\frac{n!}{e}$ .
The expected number of fixed points of a random permutation on Image may be NSFW.
Clik here to view. elements is Image may be NSFW.
Clik here to view..

As it turns out, it is possible to say substantially more about the distribution of fixed points of a random permutation. In fact, the following is true.

Theorem: As Image may be NSFW.
Clik here to view. $n \to \infty$ , the distribution of the number of fixed points of a random permutation on Image may be NSFW.
Clik here to view. elements is asymptotically Poisson with rate Image may be NSFW.
Clik here to view. $\lambda = 1$ .

A boring proof

Once we know that the number of derangements of Image may be NSFW.
Clik here to view. elements is approximately Image may be NSFW.
Clik here to view. $\frac{n!}{e}$ , we know that the number of permutations of Image may be NSFW.
Clik here to view. elements with exactly Image may be NSFW.
Clik here to view. elements is Image may be NSFW.
Clik here to view. ${n \choose k}$ times the number of derangements of Image may be NSFW.
Clik here to view. n-k elements, which is approximately

Image may be NSFW.
Clik here to view. $\displaystyle \displaystyle {n \choose k} \frac{(n-k)!}{e} = \frac{n!}{k! e}$

hence the probability that a permutation on Image may be NSFW.
Clik here to view. elements has Image may be NSFW.
Clik here to view. fixed points is approximately Image may be NSFW.
Clik here to view. $\frac{1}{k! e}$ , which is precisely the probability that a Poisson random variable with rate Image may be NSFW.
Clik here to view. $\lambda = 1$ takes the value Image may be NSFW.
Clik here to view..

A more interesting proof

For this proof we will show convergence in the sense of moments; that is, if Image may be NSFW.
Clik here to view. X_n is the random variable describing the number of fixed points of a random permutation on Image may be NSFW.
Clik here to view. elements, we will show that the moments Image may be NSFW.
Clik here to view. $\mathbb{E}(X_n^k)$ converge to the moments of a Poisson random variable Image may be NSFW.
Clik here to view. with rate Image may be NSFW.
Clik here to view. $\lambda = 1$ . First, we compute that the moment generating function of Image may be NSFW.
Clik here to view. is

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{E}(e^{tX})) = \frac{1}{e} \sum_{k \ge 0} \frac{e^{tk}}{k!} = e^{e^t - 1}$

which is the exponential generating function of the Bell numbers Image may be NSFW.
Clik here to view. B_k , which count the number of partitions of a set with Image may be NSFW.
Clik here to view. elements. We conclude that

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{E}(X^k) = B_k$

so it suffices to show that Image may be NSFW.
Clik here to view. $\mathbb{E}(X_n^k)$ converges to Image may be NSFW.
Clik here to view. B_k . Actually we will show a little more than this.

Theorem: Image may be NSFW.
Clik here to view. $\mathbb{E}(X_n^k) = B_{k, n}$ is the number of partitions of a set with Image may be NSFW.
Clik here to view. elements into at most Image may be NSFW.
Clik here to view. non-empty subsets.

Proof 1. Note that

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{E}(X_n^k) = \frac{1}{n!} \sum_{\sigma \in S_n} \text{Fix}(\sigma)^k$

where Image may be NSFW.
Clik here to view. $\text{Fix}(\sigma)^k$ is the number of fixed points of Image may be NSFW.
Clik here to view. $\sigma$ acting on the set of functions Image may be NSFW.
Clik here to view. $[k] \to [n]$ . It follows by Burnside’s lemma that the above sum is equal to the number of orbits of Image may be NSFW.
Clik here to view. S_n acting on the set of functions Image may be NSFW.
Clik here to view. $[k] \to [n]$ , which is precisely the number of partitions of Image may be NSFW.
Clik here to view. [k] into at most Image may be NSFW.
Clik here to view. non-empty subsets (by taking preimages of each element of Image may be NSFW.
Clik here to view. [n] ). Image may be NSFW.
Clik here to view. $\Box$

Proof 2. By the exponential formula,

Image may be NSFW.
Clik here to view. $\displaystyle \sum_{n \ge 0} \frac{t^n}{n!} \sum_{\sigma \in S_n} z^{\text{Fix}(\sigma)} = \exp \left( z t + \frac{t^2}{2} + ... \right)$

which simplifies to

Image may be NSFW.
Clik here to view. $\displaystyle \exp \left( (z - 1)t + \log \frac{1}{1 - t} \right) = \frac{1}{1 - t} \exp \left( (z - 1) t \right)$ .

To extract Image may be NSFW.
Clik here to view. $k^{th}$ moments, we can apply the differential operator Image may be NSFW.
Clik here to view. $\left( z \frac{\partial}{\partial z} \right)^k$ and substitute Image may be NSFW.
Clik here to view. z = 1 . Either by induction or by a combinatorial argument (see for example this math.SE question), we can show that

Image may be NSFW.
Clik here to view. $\displaystyle \left( z \frac{\partial}{\partial z} \right)^k = \sum_{r \ge 0} S(k, r) z^r \frac{\partial^r}{\partial z^r}$

where Image may be NSFW.
Clik here to view. S(k, r) are the Stirling numbers of the second kind, which count the number of partitions of a set with Image may be NSFW.
Clik here to view. elements into exactly Image may be NSFW.
Clik here to view. non-empty subsets. We conclude that

Image may be NSFW.
Clik here to view. $\displaystyle \left( z \frac{\partial}{\partial z} \right)^k \frac{1}{1 - t} \exp \left( (z - 1) t \right) = \frac{1}{1 - t} \sum_{r \ge 0} S(k, r) z^r t^r \exp \left( (z - 1) t \right)$

and substituting Image may be NSFW.
Clik here to view. z = 1 we conclude that

Image may be NSFW.
Clik here to view. $\displaystyle \sum_{n \ge 0} \frac{t^n}{n!} \sum_{\sigma \in S_n} \text{Fix}(\sigma)^k = \frac{1}{1 - t} \sum_{r \ge 0} S(k, r) t^r$ .

This gives

Image may be NSFW.
Clik here to view. $\displaystyle \frac{1}{n!} \sum_{\sigma \in S_n} \text{Fix}(\sigma)^k = \sum_{r \le n} S(k, r) = B_{k, n}$

as desired. Image may be NSFW.
Clik here to view. $\Box$

Fixed points of random permutations

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