In this problem, we are interested in enumerating permutations in $\mathcal{S}_n$ that the $i^{\textrm{th}}$ left to right maximum is $j$.
We continue our definition in
Problem 34 of
left to right maximum of a permutation $\pi=\pi_1\pi_2\cdots\pi_n\in\mathcal S_n$. Now we let $\mathrm{m}_i(\pi)$ be the $i^{\textrm{th}}$ left to right maximum of $\pi$ counting from left to right. We need to remark that if a permutation $\pi\in\mathcal S_n$ only has $k$ left to right maxima, then we take $\mathrm{m}_i(\pi)=\pi_n$ for $k < i \leq n$, i.e. the last number in $\pi$ is taken as the $i^{\textrm{th}}$ left to right maximun for $k < i \leq n$. We let $\mathrm{m}_i(\pi)=0$ for $i > n$.
For example, if we take $\pi=3145276\in\mathcal S_7$, then the set of left to right maxima of $\pi$ is $\{3,4,5,7\}$, and $\mathrm{m}_1(\pi)=3,\ \mathrm{m}_2(\pi)=4,\ \mathrm{m}_3(\pi)=5,\ \mathrm{m}_4(\pi)=7$. From $\mathrm{m}_5(\pi)$, we have $\mathrm{m}_5(\pi)=\mathrm{m}_6(\pi)=\mathrm{m}_7(\pi)=6$, and $\mathrm{m}_8(\pi)=\mathrm{m}_9(\pi)=\cdots=0$.
Let $a_{n;i,j}$ be the number of permutations $\pi\in\mathcal S_n$ such that $\mathrm{m}_i(\pi)=j$. For $i=1$, $a_{n;1,j}$ is trivial since $\mathrm{m}_1(\pi)$ is $\pi_1$, and $a_{n;1,j}$ counts the number of permutations which begins with $j$. We have $a_{n;1,j}=(n-1)!$ for all $n,j$ such that $1\leq j\leq n$.
For $i=2$, the enumeration $a_{n;2,j}$ is no longer trivial, which can be an exercise. We also have $a_{n;i,i}=(n-i)!$.
Our problem is to find a formula for $a_{n;i,j}$.