Counting Inversions

Arrange $n$ ($n \geqslant 2$) distinct numbers in some order to form a sequence $\{a_n\}$. For $1 \leqslant i < j \leqslant n$, if $a_i > a_j$, the pair $(a_i, a_j)$ is called an inversion of the sequence; the total number of inversions is the sequence's inversion number.

1. List all sequences formed from the numbers $1, 2, 3, 4$ that have exactly $2$ inversions. 2. Compute the inversion numbers of the following sequences:

1. $a_n = -2n + 19$ for $1 \leqslant n \leqslant 100$;
2. $a_n = \begin{cases}\left(\dfrac{1}{3}\right)^n, & n \text{ odd},\\[4pt] -\dfrac{n}{n+1}, & n \text{ even}\end{cases}$ for $1 \leqslant n \leqslant k$.
3. If the sequence $a_1, a_2, \ldots, a_n$ has inversion number $a$, find the inversion number of the reversed sequence $a_n, a_{n-1}, \ldots, a_1$.