The Third Problem

1000ms

262144K

Description:

You are given a permutation $$$a_1,a_2,\ldots,a_n$$$ of integers from $$$0$$$ to $$$n - 1$$$. Your task is to find how many permutations $$$b_1,b_2,\ldots,b_n$$$ are similar to permutation $$$a$$$.

Two permutations $$$a$$$ and $$$b$$$ of size $$$n$$$ are considered similar if for all intervals $$$[l,r]$$$ ($$$1 \le l \le r \le n$$$), the following condition is satisfied: $$$$$$\operatorname{MEX}([a_l,a_{l+1},\ldots,a_r])=\operatorname{MEX}([b_l,b_{l+1},\ldots,b_r]),$$$$$$ where the $$$\operatorname{MEX}$$$ of a collection of integers $$$c_1,c_2,\ldots,c_k$$$ is defined as the smallest non-negative integer $$$x$$$ which does not occur in collection $$$c$$$. For example, $$$\operatorname{MEX}([1,2,3,4,5])=0$$$, and $$$\operatorname{MEX}([0,1,2,4,5])=3$$$.

Since the total number of such permutations can be very large, you will have to print its remainder modulo $$$10^9+7$$$.

In this problem, a permutation of size $$$n$$$ is an array consisting of $$$n$$$ distinct integers from $$$0$$$ to $$$n-1$$$ in arbitrary order. For example, $$$[1,0,2,4,3]$$$ is a permutation, while $$$[0,1,1]$$$ is not, since $$$1$$$ appears twice in the array. $$$[0,1,3]$$$ is also not a permutation, since $$$n=3$$$ and there is a $$$3$$$ in the array.

Input:

Each test contains multiple test cases. The first line of input contains one integer $$$t$$$ ($$$1 \le t \le 10^4$$$) — the number of test cases. The following lines contain the descriptions of the test cases.

The first line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 10^5$$$) — the size of permutation $$$a$$$.

The second line of each test case contains $$$n$$$ distinct integers $$$a_1,a_2,\ldots,a_n$$$ ($$$0 \le a_i \lt n$$$) — the elements of permutation $$$a$$$.

It is guaranteed that the sum of $$$n$$$ across all test cases does not exceed $$$10^5$$$.

Output:

For each test case, print a single integer, the number of permutations similar to permutation $$$a$$$, taken modulo $$$10^9+7$$$.

Sample Input:

5
5
4 0 3 2 1
1
0
4
0 1 2 3
6
1 2 4 0 5 3
8
1 3 7 2 5 0 6 4

Sample Output:

Note:

For the first test case, the only permutations similar to $$$a=[4,0,3,2,1]$$$ are $$$[4,0,3,2,1]$$$ and $$$[4,0,2,3,1]$$$.

For the second and third test cases, the given permutations are only similar to themselves.

For the fourth test case, there are $$$4$$$ permutations similar to $$$a=[1,2,4,0,5,3]$$$:

$$$[1,2,4,0,5,3]$$$;
$$$[1,2,5,0,4,3]$$$;
$$$[1,4,2,0,5,3]$$$;
$$$[1,5,2,0,4,3]$$$.

Informação

Provedor Codeforces

Código CF1699C