Yet Another Array Counting Problem

2000ms

524288K

Description:

The position of the leftmost maximum on the segment $$$[l; r]$$$ of array $$$x = [x_1, x_2, \ldots, x_n]$$$ is the smallest integer $$$i$$$ such that $$$l \le i \le r$$$ and $$$x_i = \max(x_l, x_{l+1}, \ldots, x_r)$$$.

You are given an array $$$a = [a_1, a_2, \ldots, a_n]$$$ of length $$$n$$$. Find the number of integer arrays $$$b = [b_1, b_2, \ldots, b_n]$$$ of length $$$n$$$ that satisfy the following conditions:

$$$1 \le b_i \le m$$$ for all $$$1 \le i \le n$$$;
for all pairs of integers $$$1 \le l \le r \le n$$$, the position of the leftmost maximum on the segment $$$[l; r]$$$ of the array $$$b$$$ is equal to the position of the leftmost maximum on the segment $$$[l; r]$$$ of the array $$$a$$$.

Since the answer might be very large, print its remainder modulo $$$10^9+7$$$.

Input:

Each test contains multiple test cases. The first line contains a single integer $$$t$$$ ($$$1 \le t \le 10^3$$$) — the number of test cases.

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$2 \le n,m \le 2 \cdot 10^5$$$, $$$n \cdot m \le 10^6$$$).

The second line of each test case contains $$$n$$$ integers $$$a_1,a_2,\ldots,a_n$$$ ($$$1 \le a_i \le m$$$) — the array $$$a$$$.

It is guaranteed that the sum of $$$n \cdot m$$$ over all test cases doesn't exceed $$$10^6$$$.

Output:

For each test case print one integer — the number of arrays $$$b$$$ that satisfy the conditions from the statement, modulo $$$10^9+7$$$.

Sample Input:

4
3 3
1 3 2
4 2
2 2 2 2
6 9
6 9 6 9 6 9
9 100
10 40 20 20 100 60 80 60 60

Sample Output:

Note:

In the first test case, the following $$$8$$$ arrays satisfy the conditions from the statement:

$$$[1,2,1]$$$;
$$$[1,2,2]$$$;
$$$[1,3,1]$$$;
$$$[1,3,2]$$$;
$$$[1,3,3]$$$;
$$$[2,3,1]$$$;
$$$[2,3,2]$$$;
$$$[2,3,3]$$$.

In the second test case, the following $$$5$$$ arrays satisfy the conditions from the statement:

$$$[1,1,1,1]$$$;
$$$[2,1,1,1]$$$;
$$$[2,2,1,1]$$$;
$$$[2,2,2,1]$$$;
$$$[2,2,2,2]$$$.

Informação

Provedor Codeforces

Código CF1748E