A11717 | Partitions
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题目描述
You are given a set of $n$ elements indexed from $1$ to $n$ . The weight of $i$ -th element is $w_{i}$ . The weight of some subset of a given set is denoted as . The weight of some partition $R$ of a given set into $k$ subsets is  (recall that a partition of a given set is a set of its subsets such that every element of the given set belongs to exactly one subset in partition).
Calculate the sum of weights of all partitions of a given set into exactly $k$ non-empty subsets, and print it modulo $10^{9}+7$ . Two partitions are considered different iff there exist two elements $x$ and $y$ such that they belong to the same set in one of the partitions, and to different sets in another partition.
Calculate the sum of weights of all partitions of a given set into exactly $k$ non-empty subsets, and print it modulo $10^{9}+7$ . Two partitions are considered different iff there exist two elements $x$ and $y$ such that they belong to the same set in one of the partitions, and to different sets in another partition.
输入格式
The first line contains two integers $n$ and $k$ ( $1<=k<=n<=2·10^{5}$ ) — the number of elements and the number of subsets in each partition, respectively.
The second line contains $n$ integers $w_{i}$ ( $1<=w_{i}<=10^{9}$ )— weights of elements of the set.
The second line contains $n$ integers $w_{i}$ ( $1<=w_{i}<=10^{9}$ )— weights of elements of the set.
输出格式
Print one integer — the sum of weights of all partitions of a given set into $k$ non-empty subsets, taken modulo $10^{9}+7$ .
输入输出样例
输入 #1
4 2 2 3 2 3
输出 #1
160
输入 #2
5 2 1 2 3 4 5
输出 #2
645
Possible partitions in the first sample:
1. $\{\{1,2,3\},\{4\}\}$ , $W(R)=3\cdot(w_{1}+w_{2}+w_{3})+1\cdot w_{4}=24$ ;
2. $\{\{1,2,4\},\{3\}\}$ , $W(R)=26$ ;
3. $\{\{1,3,4\},\{2\}\}$ , $W(R)=24$ ;
4. $\{\{1,2\},\{3,4\}\}$ , $W(R)=2\cdot(w_{1}+w_{2})+2\cdot(w_{3}+w_{4})=20$ ;
5. $\{\{1,3\},\{2,4\}\}$ , $W(R)=20$ ;
6. $\{\{1,4\},\{2,3\}\}$ , $W(R)=20$ ;
7. $\{\{1\},\{2,3,4\}\}$ , $W(R)=26$ ;
Possible partitions in the second sample:
1. $\{\{1,2,3,4\},\{5\}\}$ , $W(R)=45$ ;
2. $\{\{1,2,3,5\},\{4\}\}$ , $W(R)=48$ ;
3. $\{\{1,2,4,5\},\{3\}\}$ , $W(R)=51$ ;
4. $\{\{1,3,4,5\},\{2\}\}$ , $W(R)=54$ ;
5. $\{\{2,3,4,5\},\{1\}\}$ , $W(R)=57$ ;
6. $\{\{1,2,3\},\{4,5\}\}$ , $W(R)=36$ ;
7. $\{\{1,2,4\},\{3,5\}\}$ , $W(R)=37$ ;
8. $\{\{1,2,5\},\{3,4\}\}$ , $W(R)=38$ ;
9. $\{\{1,3,4\},\{2,5\}\}$ , $W(R)=38$ ;
10. $\{\{1,3,5\},\{2,4\}\}$ , $W(R)=39$ ;
11. $\{\{1,4,5\},\{2,3\}\}$ , $W(R)=40$ ;
12. $\{\{2,3,4\},\{1,5\}\}$ , $W(R)=39$ ;
13. $\{\{2,3,5\},\{1,4\}\}$ , $W(R)=40$ ;
14. $\{\{2,4,5\},\{1,3\}\}$ , $W(R)=41$ ;
15. $\{\{3,4,5\},\{1,2\}\}$ , $W(R)=42$ .
1. $\{\{1,2,3\},\{4\}\}$ , $W(R)=3\cdot(w_{1}+w_{2}+w_{3})+1\cdot w_{4}=24$ ;
2. $\{\{1,2,4\},\{3\}\}$ , $W(R)=26$ ;
3. $\{\{1,3,4\},\{2\}\}$ , $W(R)=24$ ;
4. $\{\{1,2\},\{3,4\}\}$ , $W(R)=2\cdot(w_{1}+w_{2})+2\cdot(w_{3}+w_{4})=20$ ;
5. $\{\{1,3\},\{2,4\}\}$ , $W(R)=20$ ;
6. $\{\{1,4\},\{2,3\}\}$ , $W(R)=20$ ;
7. $\{\{1\},\{2,3,4\}\}$ , $W(R)=26$ ;
Possible partitions in the second sample:
1. $\{\{1,2,3,4\},\{5\}\}$ , $W(R)=45$ ;
2. $\{\{1,2,3,5\},\{4\}\}$ , $W(R)=48$ ;
3. $\{\{1,2,4,5\},\{3\}\}$ , $W(R)=51$ ;
4. $\{\{1,3,4,5\},\{2\}\}$ , $W(R)=54$ ;
5. $\{\{2,3,4,5\},\{1\}\}$ , $W(R)=57$ ;
6. $\{\{1,2,3\},\{4,5\}\}$ , $W(R)=36$ ;
7. $\{\{1,2,4\},\{3,5\}\}$ , $W(R)=37$ ;
8. $\{\{1,2,5\},\{3,4\}\}$ , $W(R)=38$ ;
9. $\{\{1,3,4\},\{2,5\}\}$ , $W(R)=38$ ;
10. $\{\{1,3,5\},\{2,4\}\}$ , $W(R)=39$ ;
11. $\{\{1,4,5\},\{2,3\}\}$ , $W(R)=40$ ;
12. $\{\{2,3,4\},\{1,5\}\}$ , $W(R)=39$ ;
13. $\{\{2,3,5\},\{1,4\}\}$ , $W(R)=40$ ;
14. $\{\{2,4,5\},\{1,3\}\}$ , $W(R)=41$ ;
15. $\{\{3,4,5\},\{1,2\}\}$ , $W(R)=42$ .
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