(noch nicht übersetzt)

A permutation $\pi$ of $\{1, \dots, n\}$ can be represented in one-line notation as $\pi(1),\ldots,\pi(n) $. If all $n!$ permutations are written in lexicographic order then $\textrm{rank}(\pi)$ is the position of $\pi$ in this 1-based list.

For example, $\text{rank}(2,1,3) = 3$ because the six permutations of $\{1, 2, 3\}$ in lexicographic order are: $$1, 2, 3\quad 1, 3, 2 \quad 2, 1, 3 \quad 2, 3, 1 \quad 3, 1, 2 \quad 3, 2, 1$$

For a positive integer $m$, we define the following permutation of $\{1, \dots, n\}$ with $n = \frac{m(m+1)}2$: $$ \begin{align} \sigma(i) &= \begin{cases} \frac{k(k-1)}2 + 1 & \textrm{if } i = \frac{k(k + 1)}2\textrm{ for }k\in\{1, \dots, m\};\\i + 1 & \textrm{otherwise};\end{cases}\\ \tau(i) &= ((10^9 + 7)i \bmod n) + 1\\ \pi(i) &= \tau^{-1}(\sigma(\tau(i))) \end{align} $$

Define $\displaystyle P(m) = \sum_{k=1}^{m!} \text{rank}(\pi^k)$.
For example, $P(2) = 4$, $P(3) = 780$ and $P(4) = 38810300$.

Find $P(100)$. Give your answer modulo $(10^9 + 7)$.