Problem 913

Row-major vs Column-major

The numbers from $1$ to $12$ can be arranged into a $3 \times 4$ matrix in either row-major or column-major order:
$R = (\begin{matrix} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 10 & 11 & 12 \end{matrix}), C = (\begin{matrix} 1 & 4 & 7 & 10 \\ 2 & 5 & 8 & 11 \\ 3 & 6 & 9 & 12 \end{matrix})$
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 $n m$ 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 $2 \leq n \leq m \leq 100$ .

行优先与列优先

$1$ 到 $12$ 这些数可以按行优先或列优先的顺序填入一个 $3 \times 4$ 矩阵中：
$R = (\begin{matrix} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 10 & 11 & 12 \end{matrix}), C = (\begin{matrix} 1 & 4 & 7 & 10 \\ 2 & 5 & 8 & 11 \\ 3 & 6 & 9 & 12 \end{matrix})$
每次交换两个元素，至少需要 $8$ 次交换才能将 $R$ 转换为 $C$ 。

令 $S (n, m)$ 为将一个包含 $1$ 到 $n m$ 的 $n \times m$ 矩阵从行优先顺序转换为列优先顺序所需的最少交换次数，因此 $S (3, 4) = 8$ 。

已知对于所有 $2 \leq n \leq m \leq 100$ ， $S (n, m)$ 之和为 $12578833$ 。

求对于所有 $2 \leq n \leq m \leq 100$ ， $S (n^{4}, m^{4})$ 之和。