# 斯特林数学习笔记

## 第一类斯特林数

$\displaystyle \begin{bmatrix}n\\m\end{bmatrix}=\begin{bmatrix}n-1\\m-1\end{bmatrix}+(n-1)*\begin{bmatrix}n-1\\m\end{bmatrix}$

### 第一类斯特林数的几条性质

#### 第一条

$\displaystyle n! = \sum_{i = 0}^{n}\begin{bmatrix}n\\i\end{bmatrix}$

$\displaystyle n! = \sum_{i = 0}^{n}\begin{bmatrix}n \\ i\end{bmatrix}$

\displaystyle \begin{aligned} \sum_{i = 0}^{n + 1}\begin{bmatrix}n + 1 \\ i\end{bmatrix} &= \begin{bmatrix}n + 1 \\ n + 1\end{bmatrix} + \sum_{i = 0}^{n}\begin{bmatrix}n + 1 \\ i\end{bmatrix}\&=\sum_{i = 0}^{n}(\begin{bmatrix}n\\i-1\end{bmatrix}+n*\begin{bmatrix}n\\i\end{bmatrix}) + \begin{bmatrix}n+1\\n+1\end{bmatrix}\&=n * \sum_{i = 0}^{n}\begin{bmatrix}n\\i\end{bmatrix}+\sum_{i=0}^{n}\begin{bmatrix}n\\i-1\end{bmatrix}+\begin{bmatrix}n+1\\n+1\end{bmatrix}\\&=n*\sum_{i=0}^{n}\begin{bmatrix}n\\i\end{bmatrix}+\sum_{i=0}^{n-1}\begin{bmatrix}n\\i\end{bmatrix}+\begin{bmatrix}n+1\\n+1\end{bmatrix}\&=(n + 1)*\sum_{i=0}^{n-1}\begin{bmatrix}n\\i\end{bmatrix}+n*\begin{bmatrix}n\\n\end{bmatrix}+\begin{bmatrix}n+1\\n+1\end{bmatrix}\&=(n + 1)*\sum_{i=0}^{n-1}\begin{bmatrix}n\\i\end{bmatrix}+n*\begin{bmatrix}n\\n\end{bmatrix}+\begin{bmatrix}n\\n\end{bmatrix}+n*\begin{bmatrix}n\\n+1\end{bmatrix}\&=(n + 1)*\sum_{i=0}^{n-1}\begin{bmatrix}n\\i\end{bmatrix}+(n +1)*\begin{bmatrix}n\\n\end{bmatrix}+0\&=(n+1)*\sum_{i=0}^{n}\begin{bmatrix}n\\i\end{bmatrix}\&=(n+1)*n!\&=(n+1)! \end{aligned}

#### 第二条

$\displaystyle x^{\underline n} = \sum_{i=0}^{n}\begin{bmatrix}n\\i\end{bmatrix}(-1)^{n-i}x^i$

$\displaystyle x^{\underline n}=\sum_{i=0}^{n}\begin{bmatrix}n\\i\end{bmatrix}(-1)^{n-i}x^i$

\displaystyle \begin{aligned} x^{\underline {n+1}}&=(x-n)x^{\underline n}\&=x\sum_{i=0}^{n}\begin{bmatrix}n\\i\end{bmatrix}(-1)^{n-i}x^i-n\sum_{i=0}^{n}\begin{bmatrix}n\\i\end{bmatrix}(-1)^{n-i}x^i\&=\sum_{i=0}^{n}\begin{bmatrix}n\\i\end{bmatrix}(-1)^{n-i}x^{i+1}-(-1)*(-1)*n\sum_{i=0}^{n}\begin{bmatrix}n\\i\end{bmatrix}(-1)^{n-i}x^i\&=\sum_{i=0}^{n+1}\begin{bmatrix}n\\i-1\end{bmatrix}(-1)^{n+1-i}x^i-\begin{bmatrix}n\\0 - 1\end{bmatrix}(-1)^{n+1}x^0+(-1)*n\sum_{i=0}^{n}\begin{bmatrix}n\\i\end{bmatrix}(-1)^{n-i}x^i\&=\sum_{i=1}^{n+1}\begin{bmatrix}n\\i - 1\end{bmatrix}(-1)^{n+1-i}x^i+n*\sum_{i = 0}^{n}\begin{bmatrix}n\\i\end{bmatrix}(-1)^{n+1-i}x^i\&=\sum_{i=1}^{n+1}\begin{bmatrix}n\\i - 1\end{bmatrix}(-1)^{n+1-i}x^i+n*\sum_{i = 0}^{n+1}\begin{bmatrix}n\\i\end{bmatrix}(-1)^{n+1-i}x^i\&=\sum_{i=1}^{n+1}(-1)^{n+1-i}x^i(\begin{bmatrix}n\\i-1\end{bmatrix}+n*\begin{bmatrix}n\\i\end{bmatrix})\&=\sum_{i=1}^{n+1}\begin{bmatrix}n+1\\i\end{bmatrix}(-1)^{n+1-i}x^i\&=\sum_{i=0}^{n+1}\begin{bmatrix}n+1\\i\end{bmatrix}(-1)^{n+1-i}x^i\\end{aligned}

#### 第三条

$\displaystyle x^{\overline n} = \sum_{i=0}^{n}\begin{bmatrix}n\\i\end{bmatrix}x^i$

$\displaystyle x^{\overline n} = \sum_{i=0}^{n}\begin{bmatrix}n\\i\end{bmatrix}x^i$

\displaystyle \begin{aligned} x^{\overline {n+1}}&= (x+n)x^{\overline n}\&=x*\sum_{i=0}^{n}\begin{bmatrix}n\\i\end{bmatrix}x^i + n * \sum_{i=0}^{n}\begin{bmatrix}n\\i\end{bmatrix}x^i\&=\sum_{i=0}^{n}\begin{bmatrix}n\\i\end{bmatrix}x^{i+1}+n*\sum_{i=0}^{n+1}\begin{bmatrix}n\\i\end{bmatrix}x^i\&=\sum_{i=1}^{n+1}\begin{bmatrix}n\\i-1\end{bmatrix}x^i+n*\sum_{i=0}^{n+1}\begin{bmatrix}n\\i\end{bmatrix}x^i\&=\sum_{i=1}^{n+1}x^i(\begin{bmatrix}n\\i-1\end{bmatrix}+n*\begin{bmatrix}n\\i\end{bmatrix})\&= \sum_{i=1}^{n+1}\begin{bmatrix}n+1\\i\end{bmatrix}x^i\&= \sum_{i=0}^{n+1}\begin{bmatrix}n+1\\i\end{bmatrix}x^i\\end{aligned}

## 第二类斯特林数

$$\begin{Bmatrix}n\\m\end{Bmatrix}$$为将$$n$$个不同的数组成$$m$$个非空集合的方案数, 即第二类斯特林数

$\displaystyle \begin{Bmatrix}n\\m\end{Bmatrix}=\begin{Bmatrix}n-1\\m-1\end{Bmatrix}+m*\begin{Bmatrix}n-1\\m\end{Bmatrix}$

### 第二类斯特林数的几条性质

#### 我所知道的唯一一条

$\displaystyle n^m=\sum_{i=0}^{m}\begin{Bmatrix}m\\i\end{Bmatrix}*C_n^i*i!$

$\displaystyle n^m=\sum_{i=0}^{m}\begin{Bmatrix}m\\i\end{Bmatrix}*C_n^i*i!$

\displaystyle \begin{aligned} n^{m+1}&=n^m*n\&=n*\sum_{i=0}^{m}\begin{Bmatrix}m\\i\end{Bmatrix}*C_n^i*i!\&=\sum_{i=0}^{m+1}n*\begin{Bmatrix}m\\i\end{Bmatrix}*C_n^i*i!\&=\sum_{i=0}^{m+1}n*\begin{Bmatrix}m\\i\end{Bmatrix}*\frac{n!}{i!*(n-i)!}*i!\&=\sum_{i=0}^{m+1}n*\begin{Bmatrix}m\\i\end{Bmatrix}*\frac{n!}{(n-i)!}\&=n!*\sum_{i=0}^{m+1}\frac{n*\begin{Bmatrix}m\\i\end{Bmatrix}}{(n-i)!}\&=n!*\sum_{i=0}^{m+1}\frac{i*\begin{Bmatrix}m\\i\end{Bmatrix}}{(n-i)!}+\frac{(n-i)*\begin{Bmatrix}m\\i\end{Bmatrix}}{(n-i)!}\&=n!*\sum_{i=0}^{m+1}\frac{i*\begin{Bmatrix}m\\i\end{Bmatrix}}{(n-i)!}+\frac{\begin{Bmatrix}m\\i\end{Bmatrix}}{(n-i-1)!}\&= n!*(\sum_{i=0}^{m+1}\frac{i*\begin{Bmatrix}m\\i\end{Bmatrix}}{(n-i)!}+\sum_{i=1}^{m+1}\frac{\begin{Bmatrix}m\\i-1\end{Bmatrix}}{(n-i)!})\&= n!*(\sum_{i=1}^{m+1}\frac{i*\begin{Bmatrix}m\\i\end{Bmatrix}}{(n-i)!}+\sum_{i=1}^{m+1}\frac{\begin{Bmatrix}m\\i-1\end{Bmatrix}}{(n-i)!})\&= n!*\sum_{i=1}^{m+1}\frac{i*\begin{Bmatrix}m\\i\end{Bmatrix}+\begin{Bmatrix}m\\i-1\end{Bmatrix}}{(n-i)!}\&= n!*\sum_{i=1}^{m+1}\frac{\begin{Bmatrix}m+1\\i\end{Bmatrix}}{(n-i)!}\&= n!*\sum_{i=0}^{m+1}\frac{\begin{Bmatrix}m+1\\i\end{Bmatrix}}{(n-i)!}\&= \sum_{i=1}^{m+1}\frac{n!*\begin{Bmatrix}m+1\\i\end{Bmatrix}}{(n-i)!}\&= \sum_{i=1}^{m+1}\frac{n!*\begin{Bmatrix}m+1\\i\end{Bmatrix}}{i!*(n-i)!}*i!\&= \sum_{i=1}^{m+1}\begin{Bmatrix}m+1\\i\end{Bmatrix}*\frac{n!}{i!*(n-i)!}*i!\&= \sum_{i=1}^{m+1}\begin{Bmatrix}m+1\\i\end{Bmatrix}*C_n^i*i!\&= \sum_{i=0}^{m+1}\begin{Bmatrix}m+1\\i\end{Bmatrix}*C_n^i*i!\\end{aligned}

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