311116: CF1936E. Yet Yet Another Permutation Problem

You are given a permutation $p$ of length $n$.

Please count the number of permutations $q$ of length $n$ which satisfy the following:

• for each $1 \le i < n$, $\max(q_1,\ldots,q_i) \neq \max(p_1,\ldots,p_i)$.

Since the answer may be large, output the answer modulo $998\,244\,353$.

Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^4$). The description of the test cases follows.

The first line of each test case contains a single integer $n$ ($2 \le n \le 2 \cdot 10^5$).

The second line of each test case contains $n$ integers $p_1, p_2, \ldots, p_n$ ($1 \le p_i \le n$). It is guaranteed that $p$ is a permutation.

It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.

For each test case, print a single integer — the answer modulo $998\,244\,353$.

6
2
2 1
3
1 2 3
3
3 1 2
4
2 4 1 3
5
3 5 1 4 2
15
6 13 2 8 7 11 1 3 9 15 4 5 12 10 14

1
3
2
4
18
424488915

In the first test case, $p = [2, 1]$. The only suitable $q$ is $[1, 2]$. Indeed, we need to satisfy the inequality $q_1 \neq p_1$. It only holds for $q = [1, 2]$.

In the second test case, $p = [1, 2, 3]$. So $q$ has to satisfy two inequalities: $q_1 \neq p_1$ and $\max(q_1, q_2) \neq \max(1, 2) = 2$. One can prove that this only holds for the following $3$ permutations:

• $q = [2, 3, 1]$: in this case $q_1 = 2 \neq 1$ and $\max(q_1, q_2) = 3 \neq 2$;
• $q = [3, 1, 2]$: in this case $q_1 = 3 \neq 1$ and $\max(q_1, q_2) = 3 \neq 2$;
• $q = [3, 2, 1]$: in this case $q_1 = 3 \neq 1$ and $\max(q_1, q_2) = 3 \neq 2$.