Question 10
Call a permutation $a_1,$ $a_2,$ $\ldots,$ $a_n$ of the integers $1,$ $2,$ $\ldots,$ $n$ *quasi-increasing* if $a_k \le a_{k+1} + 2$ for each $1 \le k \le n - 1.$ For example, $53421$ and $14253$ are quasi-increasing permutations of the integers $1,$ $2,$ $3,$ $4,$ $5,$ but $45123$ is not. Find the number of quasi-increasing permutations of the integers $1,$ $2,$ $\ldots,$ $7.$