Eccoci.

The numbers from $1$ to $12$ can be arranged into a $3 \times 4$ matrix in either row-major or column-major order: $$R=\begin{pmatrix} 1 & 2 & 3 & 4\\ 5 & 6 & 7 & 8\\ 9 & 10 & 11 & 12\end{pmatrix}, C=\begin{pmatrix} 1 & 4 & 7 & 10\\ 2 & 5 & 8 & 11\\ 3 & 6 & 9 & 12\end{pmatrix}$$ By swapping two entries at a time, at least $8$ swaps are needed to transform $R$ to $C$.

Let $S(n, m)$ be the minimal number of swaps needed to transform an $n\times m$ matrix of $1$ to $nm$ from row-major order to column-major order. Thus $S(3, 4) = 8$.

You are given that the sum of $S(n, m)$ for $2 \leq n \leq m \leq 100$ is $12578833$.

Find the sum of $S(n^4, m^4)$ for $2 \leq n \leq m \leq 100$.