Question 172
Find the number of permutations $\left(a_{1}, a_{2}, \ldots, a_{2016}\right)$ of the first $2016$ positive integers satisfying the following two conditions:
1. $a_{i+1}-a_{i} \leq 1$ for all $i=1,2,3, \ldots, 2015$.
2. There are exactly two indices $i \lt j$ with $1 \leq i \lt j \leq 2016$ such that $a_{i}=i$ and $a_{j}=j$.
1. $a_{i+1}-a_{i} \leq 1$ for all $i=1,2,3, \ldots, 2015$.
2. There are exactly two indices $i \lt j$ with $1 \leq i \lt j \leq 2016$ such that $a_{i}=i$ and $a_{j}=j$.