Derrangments of an -set, Combinatorial proofs, and EGFs

Question

Let $D_n$ be the number of derrangements of an n-set. $D_o=1$. Give a combinatorial proof that $D_n = (n-1)(D_{n-1}+D_{n-2})$ for $n \geq 2$ and find an EGF of {$D_n$}$_{n \geq 0}$.

Answer 1

A derangement $\pi$ of $[n]=\{1,\dots,n\}$ must send $n$ to one of the integers in $[n-1]$. Suppose that $\pi(n)=k$. If $\pi(k)=n$, $\pi$ must restrict to a derangement of $[n-1]\setminus\{k\}$, and there are $D_{n-2}$ of those. Otherwise there is some $m\in[n-1]\setminus\{k\}$ such that $\pi(m)=n$. Let

$$\pi':[n-1]\to[n-1]:i\mapsto\begin{cases} \pi(i),&\text{if }i\ne m\\ k,&\text{if }i=m\;. \end{cases}$$

Show that $\pi'$ is a derangement of $[n-1]$ and that every derangement of $[n-1]$ can be obtained in this way from a derangement $\pi$ of $n$ such that $\pi(n)=k$. Since there are $n-1$ choices for $k$, conclude that $D_n=(n-1)(D_{n-1}+D_{n-2})$.

For the exponential generating function see this earlier answer.