CodeForces 785 D Anton and School - 2 范德蒙恒等式

Anton and School - 2

题解：

枚举每个左括号作为必选的。

那么方案数就应该是下面的 1 ， 然后不断化简， 通过范德蒙恒等式 ， 可以将其化为一个组合数。

代码：

#include<bits/stdc++.h>

using namespace std;

#define Fopen freopen("_in.txt","r",stdin); freopen("_out.txt","w",stdout);

#define LL long long

#define ULL unsigned LL

#define fi first

#define se second

#define pb push_back

#define lson l,m,rt<<1

#define rson m+1,r,rt<<1|1

#define lch(x) tr[x].son[0]

#define rch(x) tr[x].son[1]

#define max3(a,b,c) max(a,max(b,c))

#define min3(a,b,c) min(a,min(b,c))

typedef pair<int,int> pll;

const int inf = 0x3f3f3f3f;

const int _inf = 0xc0c0c0c0;

const LL INF = 0x3f3f3f3f3f3f3f3f;

const LL _INF = 0xc0c0c0c0c0c0c0c0;

const LL mod =  (int)1e9+;

const int N = 2e5 + ;

int F[N], Finv[N], inv[N];/// F是阶层 Finv是逆元的阶层

void init(){

    inv[] = ;

    for(int i = ; i < N; i++)

        inv[i] = (mod - mod/i) * 1ll * inv[mod % i] % mod;

    F[] = Finv[] = ;

    for(int i = ; i < N; i++){

        F[i] = F[i-] * 1ll * i % mod;

        Finv[i] = Finv[i-] * 1ll * inv[i] % mod;

    }

}

int comb(int n, int m){ /// C(n,m)

    if(m <  || m > n) return ;

    return F[n] * 1ll * Finv[n-m] % mod * Finv[m] % mod;

}

char s[N];

int l[N], r[N];

int main(){

    scanf("%s", s+);

    int n = strlen(s+);

    for(int i = ; i <= n; ++i){

        if(s[i] == '(') l[i]++;

        l[i] += l[i-];

    }

    for(int i = n; i >= ; --i){

        if(s[i] == ')') r[i]++;

        r[i] += r[i+];

    }

    LL ans = ;

    init();

    for(int i = ; i <= n; ++i){

        if(s[i] == '('){

            ans = (ans + comb(l[i]-+r[i], l[i]))%mod;

        }

    }

    cout << ans << endl;

    return ;

}