I'm having trouble finding some elementary results on the following. Let $Y$ be a standard Young Tableau of shape $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_n)$ with $N:=\sum_{i=1}^n\lambda_i$.
My question is, for a given element $k$, what can be said about the location of $k$ in terms of bounds on its $(i,j)$ coordinates in the tableau? For example, 1 must be at $(1,1)$, 2 can be at $(2,1)$ or $(1,2)$, 3 can be at $(2,1),(3,1),(1,2),(1,3)$ and so on. What I'm looking for is some crude statement of the form: $k$ lies in the region bounded by intervals $[(i_1,j_1),(i_1,j_1')],\ldots,[(i_n,j_n),(i_n,j_n')]$ where the intervals are defined between the two bracketed coordinates. References would be highly appreciated!
The lower bound is $k\ge ij$, as we need to put the numbers $1,\ldots, k-1$ "below" $k$. The upper bound depends on the tableau. We cannot place $k$ at $(i,j)$ if we cannot place $k-1$ elements in the columns less than $i$ and the rows less than $j$. This translates to $$k \leq \sum_{k=1}^{i-1}\lambda_k + \sum_{l=1}^{j-1}\lambda'_l - (i-1)(j-1)$$ where $\lambda'_1,\lambda'_2,\ldots$ describe the conjugate tableau. (I forget the standard notation for this.) We need to subtract $(i-1)(j-1)$ to avoid double counting.