CF1909F2.Small Permutation Problem (Hard Version)

Andy Tunstall - MiniBoss

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In the easy version, the $a_i$ are in the range $[0, n]$ ; in the hard version, the $a_i$ are in the range $[-1, n]$ and the definition of good permutation is slightly different. You can make hacks only if all versions of the problem are solved.

You are given an integer $n$ and an array $a_1, a_2, \dots, a_n$ of integers in the range $[-1, n]$ .

A permutation $p_1, p_2, \dots, p_n$ of $[1, 2, \dots, n]$ is good if, for each $i$ , the following condition is true:

• if $a_i \neq -1$ , the number of values $\leq i$ in $[p_1, p_2, \dots, p_i]$ is exactly $a_i$ .

Count the good permutations of $[1, 2, \dots, n]$ , 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 $n$ ( $1 \leq n \leq 2 \cdot 10^5$ ) — the length of the array $a$ .

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ( $-1 \le a_i \le n$ ), which describe the conditions for a good permutation.

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

For each test case, output a single line containing the number of good permutations, modulo $998\,244\,353$ .

In the first test case, all the permutations of length $5$ are good, so there are $120$ good permutations.

In the second test case, the only good permutation is $[1, 2, 3, 4, 5]$ .

In the third test case, there are $4$ good permutations: $[2, 1, 5, 6, 3, 4]$ , $[2, 1, 5, 6, 4, 3]$ , $[2, 1, 6, 5, 3, 4]$ , $[2, 1, 6, 5, 4, 3]$ . For example, $[2, 1, 5, 6, 3, 4]$ is good because:

• $a_1 = 0$ , and there are $0$ values $\leq 1$ in $[p_1] = [2]$ ;
• $a_2 = 2$ , and there are $2$ values $\leq 2$ in $[p_1, p_2] = [2, 1]$ ;
• $a_3 = 2$ , and there are $2$ values $\leq 3$ in $[p_1, p_2, p_3] = [2, 1, 5]$ ;
• $a_4 = 2$ , and there are $2$ values $\leq 4$ in $[p_1, p_2, p_3, p_4] = [2, 1, 5, 6]$ ;
• $a_5 = -1$ , so there are no restrictions on $[p_1, p_2, p_3, p_4, p_5]$ ;
• $a_6 = -1$ , so there are no restrictions on $[p_1, p_2, p_3, p_4, p_5, p_6]$ .