CF1917D.Yet Another Inversions Problem

You are given a permutation $p_0, p_1, \ldots, p_{n-1}$ of odd integers from $1$ to $2n-1$ and a permutation $q_0, q_1, \ldots, q_{k-1}$ of integers from $0$ to $k-1$ .

An array $a_0, a_1, \ldots, a_{nk-1}$ of length $nk$ is defined as follows:

$a_{i \cdot k+j}=p_i \cdot 2^{q_j}$ for all $0 \le i < n$ and all $0 \le j < k$ For example, if $p = [3, 5, 1]$ and $q = [0, 1]$ , then $a = [3, 6, 5, 10, 1, 2]$ .

Note that all arrays in the statement are zero-indexed. Note that each element of the array $a$ is uniquely determined.

Find the number of inversions in the array $a$ . Since this number can be very large, you should find only its remainder modulo $998\,244\,353$ .

An inversion in array $a$ is a pair $(i, j)$ ( $0 \le i < j < nk$ ) such that $a_i > a_j$ .

The first line contains a single integer $t$ ( $1 \le t \le 10^4$ ) — the number of test cases.

The first line of each test case contains two integers $n$ and $k$ ( $1 \le n, k \le 2 \cdot 10^5$ ) — the lengths of arrays $p$ and $q$ .

The second line of each test case contains $n$ distinct integers $p_0, p_1, \ldots, p_{n-1}$ ( $1 \le p_i \le 2n-1$ , $p_i$ is odd) — the array $p$ .

The third line of each test case contains $k$ distinct integers $q_0, q_1, \ldots, q_{k-1}$ ( $0 \le q_i < k$ ) — the array $q$ .

It is guaranteed that the sum of $n$ over all test cases doesn't exceed $2 \cdot 10^5$ and the sum of $k$ over all test cases doesn't exceed $2 \cdot 10^5$ .

For each test case, output one integer: the number of inversions in array $a$ modulo $998\,244\,353$ .

In the first test case, array $a$ is equal to $[3, 6, 5, 10, 1, 2]$ . There are $9$ inversions in it: $(0, 4)$ , $(0, 5)$ , $(1, 2)$ , $(1, 4)$ , $(1, 5)$ , $(2, 4)$ , $(2, 5)$ , $(3, 4)$ , $(3, 5)$ . Note that these are pairs $(i, j)$ such that $i < j$ and $a_i > a_j$ .

In the second test case, array $a$ is equal to $[8, 4, 1, 2, 24, 12, 3, 6, 40, 20, 5, 10]$ . There are $25$ inversions in it.

In the third test case, array $a$ is equal to $[1, 2, 4, 8, 16]$ . There are no inversions in it.