For a partition $\lambda=(\lambda_1\geq \lambda_2\geq\ldots\geq \lambda_k)$ of $n$, let the set of standard Young tableau of shape $\lambda$ be denoted by $SYT(\lambda)$ with boxes at $(i,j)$ denoted by $B_{ij}$ taking in distinct values in $\{1,2,\ldots,n\}$. Also let $f_\lambda:=|SYT(\lambda)|$. Finally, let $N_{ij}(k)$ be the set of young tableau of shape $\lambda$ with $B_{ij}=k$. We will exclusively be talking about boxes on the right edge of the Young tableaux (boxes that have no boxes directly south or east of them). We will call these corner boxes. In the diagram below, 8,9 and 10 are corner boxes. On the other hand, 5 and 6 are not corners.

The usual branching rule says that

$$f_\lambda=\sum_{\mu\rightarrow\lambda}f_\mu,$$

where the sum is taken over all partitions $\mu$ of $n-1$ that are contained ($\rightarrow$) in $\lambda$. It is easy to see that when $(i,j)$ is a corner box, $N_{ij}(n)=f_{\lambda- (i,j)}$, the number of SYT of shape $\lambda$ minus the corner box in question. This kind of reasoning can be extended to $N_{ij}(n-1),N(n-2)$. Unfortunately, $N_{ij}(n-k)$ becomes exceeding difficult to compute for $k>2$.

I would like to ask what is known about $N_{ij}(k)$, when $(i,j)$ is a corner box. Specifically, are there any recurrences that these satisfy? Are they at all related to coefficients of certain weighted hook-walk algorithms? Have they been considered in any enumeration problems? Does this question have an answer for certain fixed nontrivial shapes $\lambda$?