Close Tuples (hard version)

4000ms

262144K

Description:

This is the hard version of this problem. The only difference between the easy and hard versions is the constraints on $$$k$$$ and $$$m$$$. In this version of the problem, you need to output the answer by modulo $$$10^9+7$$$.

You are given a sequence $$$a$$$ of length $$$n$$$ consisting of integers from $$$1$$$ to $$$n$$$. The sequence may contain duplicates (i.e. some elements can be equal).

Find the number of tuples of $$$m$$$ elements such that the maximum number in the tuple differs from the minimum by no more than $$$k$$$. Formally, you need to find the number of tuples of $$$m$$$ indices $$$i_1 < i_2 < \ldots < i_m$$$, such that

$$$$$$\max(a_{i_1}, a_{i_2}, \ldots, a_{i_m}) - \min(a_{i_1}, a_{i_2}, \ldots, a_{i_m}) \le k.$$$$$$

For example, if $$$n=4$$$, $$$m=3$$$, $$$k=2$$$, $$$a=[1,2,4,3]$$$, then there are two such triples ($$$i=1, j=2, z=4$$$ and $$$i=2, j=3, z=4$$$). If $$$n=4$$$, $$$m=2$$$, $$$k=1$$$, $$$a=[1,1,1,1]$$$, then all six possible pairs are suitable.

As the result can be very large, you should print the value modulo $$$10^9 + 7$$$ (the remainder when divided by $$$10^9 + 7$$$).

Input:

The first line contains a single integer $$$t$$$ ($$$1 \le t \le 2 \cdot 10^5$$$) — the number of test cases. Then $$$t$$$ test cases follow.

The first line of each test case contains three integers $$$n$$$, $$$m$$$, $$$k$$$ ($$$1 \le n \le 2 \cdot 10^5$$$, $$$1 \le m \le 100$$$, $$$1 \le k \le n$$$) — the length of the sequence $$$a$$$, number of elements in the tuples and the maximum difference of elements in the tuple.

The next line contains $$$n$$$ integers $$$a_1, a_2,\ldots, a_n$$$ ($$$1 \le a_i \le n$$$) — the sequence $$$a$$$.

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

Output:

Output $$$t$$$ answers to the given test cases. Each answer is the required number of tuples of $$$m$$$ elements modulo $$$10^9 + 7$$$, such that the maximum value in the tuple differs from the minimum by no more than $$$k$$$.

Sample Input:

4
4 3 2
1 2 4 3
4 2 1
1 1 1 1
1 1 1
1
10 4 3
5 6 1 3 2 9 8 1 2 4

Sample Output:

Informação

Provedor Codeforces

Código CF1462E2