N个数依次入栈，出栈顺序有多少种？

//模拟过程如下，dfs来填充入栈和出栈的标志
#include<iostream> 
#include<cstring>
#include<cstdio>
#include<stack>
#define N 100
using namespace std;

int a[N];
int p[2*N];
int pre[2*N];
int used[2*N];
int index[2*N];//如果a[i]==1，那么表示第index[i]个数入栈 
int n;
int cnt;//记录多少个出栈的序列 

int C(int n){
    int ans=1;
    int m = 2*n;
    int nn = n;
    for(int i=1; i<=n; ++i){
        ans *= m--;
        ans /= i;
    }
    return ans/(nn+1);
}

void solve(){
    memset(pre, 0, sizeof(pre));
    memset(used, 0, sizeof(used));
    int prek = 0;
    for(int i=1; i<=2*n; ++i){
        if(a[i] == 1){
            pre[i] = prek;
            prek = i;
        } else {
            int ii = prek;
            while(used[ii]) 
                ii = pre[ii];
            cout<<p[index[ii]]<<" ";//出栈 
            used[ii] = 1;
            pre[prek] = pre[ii];
        }
    }
    cout<<endl;
}

void solve1(){
    stack<int>s;
    int k=1;
    for(int i=1; i<=2*n; ++i){
        if(a[i] == 1)
            s.push(p[k++]);
        else {
            cout<<s.top()<<" ";
            s.pop();
        }
    }
    cout<<endl;
}

void dfs(int k, int cnt0, int cnt1){
    if(k>2*n){
        //solve(); 
        solve1(); 
        ++cnt; 
        return ;
    }
    if(cnt1<n){//入栈 
        a[k] = 1;
        index[k] = cnt1+1;//第几个数入栈 
        dfs(k+1, cnt0, cnt1+1);
    }
    if(cnt0<cnt1){//出栈 
        a[k] = 0;
        dfs(k+1, cnt0+1, cnt1);
    }
}

int main(){
    cin>>n;
    for(int i=1; i<=n; ++i) 
        cin>>p[i];
    dfs(1, 0, 0);
    cout<<endl<<"模拟个数:"<<cnt<<endl;
    cout<<"公式个数:"<<C(n)<<endl;
    return 0;
}