A11278. 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 $998\,244\,353$ .

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 $\max(q_1, q_2) = 3 \neq 2$ .