> For the complete documentation index, see [llms.txt](https://mqjyl2012.gitbook.io/algorithm/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mqjyl2012.gitbook.io/algorithm/leetcode/permutation-and-combination.md).

# 排列 & 组合

## :pencil2: 1、排列

### 1.1、全排列

### 1.2、全排列（有重复元素）

### 1.3、[第k个排列](https://leetcode-cn.com/problems/permutation-sequence/)

给出集合 `[1,2,3,…,n]`，其所有元素共有 `n!` 种排列。按从小到大的顺序列出所有排列情况，返回第 k 个排列。

说明：给定 n 的范围是 `[1, 9]`。 给定 k 的范围是`[1, n!]`。

> 思路：确定了 $$a\_1$$ 后，这些排列从编号 $$(a\_1 - 1)\cdot(n - 1)!$$ 开始，到 $$a\_1\cdot(n - 1)!$$ 结束，总计 $$(n - 1)!$$ 个，因此第 k 个排列实际上就对应着这其中的第 $$k \prime =(k−1)mod(n−1)!+1$$ 个排列。这样一来，我们就把原问题转化成了一个完全相同但规模减少 1 的子问题：求 $$\[1, n] \backslash a\_1$$ 这 $$n - 1$$ 个元素组成的排列中，第 $$k\prime$$ 小的排列。

```cpp
string getPermutation(int n, int k) {
    vector<int> factorials = {1, 1, 2, 6, 24, 120, 720, 5040, 40320};
    vector<int> nums(n, 0);
    iota(std::begin(nums), std::end(nums), 1);
    string ret;
    k --; // 从第 0 个开始
    for(int i = n - 1; i >= 0; i --){
        auto it = nums.begin() + k / factorials[i];
        ret += ('0' + *it);
        nums.erase(it);
        k %= factorials[i];
    }
    return ret;
}
```

> 对于给定的排列 $$a\_1, a\_2, \cdots, a\_n$$，你能求出 k 吗？
>
> 解答： $$k = \left( \sum\_{i=1}^n \textit{order}(a\_i) \cdot (n-i)! \right) + 1$$ ，其中 $$\text{order}(a\_i)$$ 表示 $$a\_{i+1}, \cdots, a\_n$$ 中小于 $$a\_i$$ 的元素个数。

## :pencil2: 2、组合

## :pencil2: 3、子集


---

# Agent Instructions
This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com.

## Querying This Documentation
If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter, and the optional `goal` query parameter:

```
GET https://mqjyl2012.gitbook.io/algorithm/leetcode/permutation-and-combination.md?ask=<question>&goal=<endgoal>
```

`ask` is the immediate question: it should be specific, self-contained, and written in natural language.
`goal` is optional and describes the broader end goal you are ultimately trying to accomplish on behalf of the user. GitBook uses it to tailor the answer towards what is most useful for that goal.

The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.