递归与排列组合问题

指数型枚举

#include <iostream>
using namespace std;

int n, num[15];

void print(int cnt) {
    cout << num[1];
    for (int i = 2; i <= cnt; i++) {
        cout << " " << num[i];
    }
    cout << endl;
}

//从第几个数字开始，更改第几位的数字
void func(int start, int number) {
    for (int i = start; i <= n; i++) {
        num[number] = i;
        print(number);
        func(i + 1, number + 1);
    }
}

int main() {
    cin >> n;
    func(1, 1);
    return 0;
}

结果

输入: 4
1
1 2
1 2 3
1 2 3 4
1 2 4
1 3
1 3 4
1 4
2
2 3
2 3 4
2 4
3
3 4
4

组合型枚举

#include <iostream>
using namespace std;

int n, m, num[15];

void print(int cnt) {
    cout << num[1];
    for (int i = 2; i <= cnt; i++) {
        cout << " " << num[i];
    }
    cout << endl;
}

//从第几个数字开始，更改第几位的数字
void func(int start, int number) {
    if (number == m + 1) {
        print(m);
        return;
    }
    for (int i = start; i <= n; i++) {
        num[number] = i;
        func(i + 1, number + 1);
    }
}

int main() {
    cin >> n >> m;
    func(1, 1);
    return 0;
}

结果

输入: 5 3
1 2 3
1 2 4
1 2 5
1 3 4
1 3 5
1 4 5
2 3 4
2 3 5
2 4 5
3 4 5

排列型枚举

#include <iostream>
using namespace std;

int n, m, num[15], mark[15];

void print(int cnt) {
    cout << num[1];
    for (int i = 2; i <= cnt; i++) {
        cout << " " << num[i];
    }
    cout << endl;
}

//更改第几位的数字
void func(int number) {
    if (number == m + 1) {
        print(m);
        return;
    }
    for (int i = 1; i <= n; i++) {
        if (mark[i] == 1) continue;
        num[number] = i;
        mark[i] = 1;
        func(number + 1);
        mark[i] = 0;
    }
}

int main() {
    cin >> n >> m;
    func(1);
    return 0;
}

结果

输入: 3 3
1 2 3
1 3 2
2 1 3
2 3 1
3 1 2
3 2 1

（总结）递归二要素：

1. 递归边界
2. 递归函数的意义！！！