Derangements $p$ of $1,2,\dots,n,n+1$ such that $n+1$ doesn't go to $n$

Question

Recall that the number or Derangements of $1,2,\dots,n$ is a permutation $p$ such that $p(i) \neq i$ for all $i$. We can express it with the recurrence $D_n=(n-1)(D_{n-1}+D_{n-2})$ or by the closed formula $$D_n =\sum_{i=0}^n (-1)^i \frac{n!}{i!}$$

Now we consider the number of permutations of $1,2,\dots, n,n+1$ such that for all $1\leq i \leq n$ (not $n+1$), $p(i)\neq i$, but also $p(n+1) \neq n$. We need to express this number using $D_n$s.

I was thinking of first counting the number of permutations without the additional condition $p(n+1) \neq n$ and received using inclusion-exclusion

$$\sum_{r=0}^n (-1)^r \binom{n}{r}(n+1-r)! = \sum_{r=0}^n (-1)^r \frac{n!}{r!}(n+1-r) =$$ $$\sum_{r=0}^n (-1)^r \frac{(n+1)!}{r!}- \sum_{r=1}^n (-1)^r \frac{n!}{(r-1)!}=$$ $$\sum_{r=0}^n (-1)^r \frac{(n+1)!}{r!}- n\sum_{r=0}^{n-1} (-1)^{r+1} \frac{(n-1)!}{r!}=$$ $$=D_{n+1}+nD_{n-1}$$

Now we need to subtract from this the number of permutations when $p(n+1)=n$, which I wasn't able to count.

Thanks in advance

Answer 1

These kind of problems (permutations with forbidden positions) can be elegantly solved using rook numbers/polynomials:

Let $P \subseteq \{1, \ldots, n\}^2$ be the diagram of forbidden positions. The number of permutations $p \in S_n$ such that $(i,p(i)) \notin P$ for all $i = 1, \ldots, n$ is given by $$\sum_{k=0}^n (-1)^k(n-k)!r_k$$ where $r_k$ is the number of ways to place $k$ nonattacking rooks on the diagram $P$.

In our case the diagram e.g. for $n=4$ (the board is then $5 \times 5$) is given by:

$0$ rooks can be placed in one way on $P$. To place $k$ rooks on $P$, one can place a rook on one of the two positions in the last column of $P$ in $2$ ways and then place $k-1$ rooks on $n-1$ remaining positions, or one can just place all $k$ rooks in the first $n-1$ columns, ignoring the last one. Hence $$r_k = \begin{cases} 1, \text{ if }k=0\\ 2{n-1 \choose k-1} + {n-1 \choose k}, \text{ if }k \ge 1 \end{cases} = \begin{cases} 1, \text{ if }k=0\\ {n \choose k} + {n-1 \choose k-1}, \text{ if }k \ge 1 \end{cases}$$

Therefore the result is \begin{align} \sum_{k=0}^n (-1)^k(n-k)!r_k &= \sum_{k=0}^n (-1)^k(n-k)!{n \choose k} + n! + \sum_{k=1}^n (-1)^k(n-k)!{n-1 \choose k-1} \\ &= D_n + n! - \sum_{j=0}^{n-1} (-1)^j((n-1)-j)!{n-1 \choose j} \\ &= D_n + n! - D_{n-1} \end{align}

Answer 2

Call a permutation $\sigma$ of $\{1,\ldots,n+1\}$ good if $\sigma(i)\neq i$ for all $1\leq i\leq n$ and and $\sigma(n+1)\neq n$.

Let $\sigma$ be a good permutation, and let $a:=\sigma(n+1)$ and $b:=\sigma^{-1}(n+1)$, so that $a\neq n$. If $a\neq b$ then the permutation $(a\ n+1)\sigma$ fixes $n+1$ and is a derangement of $\{1,\ldots,n\}$. If $a=b$ then $(a\ n+1)\sigma$ fixes $a$ and $n+1$, and is a derangement of $\{1,\ldots,n\}\setminus\{a\}$.

Conversely, for any derangement $\sigma$ of $\{1,\ldots,n\}$ and any $a\in\{1,\ldots,n+1\}$ with $a\neq n$ the composition $(a\ n+1)\sigma$ is good. Also, for any $a\in\{1,\ldots,n-1\}$ and any derangement $\sigma$ of $\{1,\ldots,n\}\setminus\{a\}$ the composition $(a\ n+1)\sigma$ is good.

This shows that the number of good permutations is $nD_n+(n-1)D_{n-1}$, where $D_m$ denotes the number of derangements of $\{1,\ldots,m\}$.