310918: CF1909F2. Small Permutation Problem (Hard Version)

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 $998\,244\,353$.

10
5
-1 -1 -1 -1 -1
5
1 2 3 4 5
6
0 2 2 2 -1 -1
6
-1 -1 -1 -1 -1 5
6
-1 -1 3 2 -1 -1
15
0 0 -1 -1 -1 2 2 -1 -1 -1 -1 9 11 13 15
6
0 2 2 2 4 6
6
0 1 3 4 5 5
6
1 2 3 2 4 6
15
0 0 1 1 1 2 3 4 5 6 7 9 11 13 15

120
1
4
0
0
494403526
4
0
0
532305727

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 $[p_1, p_2, p_3, p_4, p_5, p_6]$.