CF785D Anton and School - 2（范德蒙德行列式卷积）

Posted 2021-10-02 goto_1600

题意：
给定含有(和)的字符串，求带有((()))类似的子序列的方案数（左括号等于右括号）
思路：
很容易想到枚举每个左括号的最后一个位置，cnt代1~i-1中(的数量，cnt2代表后缀的）的数量那么显然答案就是
${\sum }_{i=1}^n[s[i]==(]{\sum }_{j=0}^{min(cnt,cnt2-1)}C(cnt,cnt-j)\ast C(cnt2,j+1)$ ，那么这样是$n^2$的，看了题解，发现是范德蒙德卷积，范德蒙德卷积定义，它是专门来处理${\sum }_{i=0}^{i=k}C(a,i)\ast C(b,k-i)=C(a+b,k)$感性的理解就是从a集合和b集合中拿出k个数总共有多少种方案。

// Problem: CF785D Anton and School - 2
// Contest: Luogu
// URL: https://www.luogu.com.cn/problem/CF785D
// Memory Limit: 250 MB
// Time Limit: 2000 ms
// 
// Powered by CP Editor (https://cpeditor.org)

//#pragma GCC target("avx")
//#pragma GCC optimize(2)
//#pragma GCC optimize(3)
//#pragma GCC optimize("Ofast")
// created by myq 
#include<iostream>
#include<cstdlib>
#include<string>
#include<cstring>
#include<cstdio>
#include<algorithm>
#include<climits>
#include<cmath>
#include<cctype>
#include<stack>
#include<queue>
#include<list>
#include<vector>
#include<set>
#include<map>
#include<sstream>
#include<unordered_map>
#include<unordered_set>
using namespace std;
typedef long long ll;
#define x first
#define y second
typedef pair<int,int> pii;
const int N = 400010;
const int mod=1e9+7;
inline int read()
{
	int res=0;
	int f=1;
	char c=getchar();
	while(c>'9' ||c<'0')
	{
		if(c=='-')	f=-1;
		c=getchar();
	}
	while(c>='0'&&c<='9')
	{
		res=(res<<3)+(res<<1)+c-'0';
		c=getchar(); 
	}
	return res;
 } 
 #define int long long
 int fact[N],infact[N];
 int qmi(int a,int b){
 	int res=1;
 	while(b){
 		if(b&1)	res=res*a%mod;
 		b>>=1;
 		a=a*a%mod;
 	}
 	return res;
 }
 int c(int a,int b){
 	if(a<b)	return 0;
 	return fact[a]*infact[b]%mod*infact[a-b]%mod;
 }
signed main() 
{ 
	fact[0]=1;
	infact[0]=1;
	for(int i=1;i<N;i++)	fact[i]=fact[i-1]*i%mod,infact[i]=qmi(fact[i],mod-2);
	string s;
	cin>>s;
	int n=s.size();
	int cnt=0;
	int cnt2=0;
	for(int i=0;i<s.size();i++){
		if(s[i]==')')	cnt2++;
	}
	int res=0;
	for(int i=0;i<s.size();i++){
		if(s[i]=='('){
				res=(res+c(cnt+cnt2,cnt+1))%mod;

			cnt++;
		}
		else{
			cnt2--;
		}
	}
	cout<<res<<endl;
	return 0;
	
}
/**
* In every life we have some trouble
* When you worry you make it double
* Don't worry,be happy.
**/