Given a Young diagram $Y$, for each row $R$ choose a permutation $\sigma_R$ of $\{1,\dots, |R|\}$, where $|R|$ is the size of row $R$. Let $\sigma_R(i)$ be the “row coordinate” of the $i$th cell in row $R$.
And for each column $C$ choose a permutation $\tau_C$ of $\{1,\dots, |C|\}$, and let $\tau(j)$ be the “column coordinate” of the $j$th cell in column $C$.
The goal of choosing those permutations is to make the sum of the row coordinate and column coordinate of each cell as small as possible.
For example, when the Young diagram is a square, the sum can be as small as $|R|+1=|C|+1$ for every cell. What about other types of Young diagrams? Is there any result about this type of question?
Edit: to be more concrete, assume the diagram has rows $R_1,\dots, R_m$ (whose lengths are decreasing) and columns $C_1,\dots, C_n$ (whose lengths are decreasing). For the cell $(i,j)\in R_i\cap C_j$, let $x(i,j),y(i,j)$ be its row and column coordinates, respectively. (so $x(i,j)=\sigma_{R_i}(j)$ and $y(i,j)=\tau_{C_j}(i)$.)
Let $$t_i=\max_{(i,j)\in R_i}(x(i,j)+y(i,j)).$$
The goal is to minimize $$||t||_{\ell_1}=\sum_{i=1}^m t_i.$$
A trivial bound is $$ \sum_{i=1}^m(|R_i|+1)\le ||t|| \le \sum_{i=1}^m(|R_1|+1).$$
Any equivalent condition (or some non-trivial condition besides all the rows having the same length and $|R_1|\ge |C_1|$) that in which case, the lower bound can be attained?
