$$F(n)=\sum\limits_{d|n}f(d)\Rightarrow f(n)=\sum\limits_{d|n}\mu(\frac n d)F(d)$$
$$F(n)=\sum\limits_{n|d}f(d)\Rightarrow f(n)=\sum\limits_{n|d}\mu(\frac d n)F(d)$$
$$f(n)=\sum\limits_{i=0}^n(-1)^i\binom{n}{i}g(i)\Rightarrow g(n)=\sum\limits_{i=0}^n(-1)^i\binom{n}{i}f(i)$$
$$f(n)=\sum\limits_{i=0}^n\binom{n}{i}g(i)\Rightarrow g(n)=\sum\limits_{i=0}^n(-1)^{n-i}\binom{n}{i}f(i)$$
$$f(k)=\sum\limits_{i=k}^n\binom{i}{k}g(i)\Rightarrow g(k)=\sum\limits_{i=k}^n(-1)^{i-k}\binom{i}{k}f(i)$$
$$f(n)=\sum\limits_{i=0}^n\begin{Bmatrix}n \\i\end{Bmatrix}g(i)\Rightarrow g(n)=\sum\limits_{i=0}^n(-1)^{n-i}\begin{bmatrix}n \\i\end{bmatrix}f(i)$$
$$f(k)=\sum\limits_{i=k}^n\begin{Bmatrix}i \\k\end{Bmatrix}g(i)\Rightarrow g(n)=\sum\limits_{i=k}^n(-1)^{i-k}\begin{bmatrix}i \\k\end{bmatrix}f(i)$$
$$\max\{S\}=\sum\limits_{T\subseteq S}(-1)^{|T|-1}\min\{T\}$$
$$\min\{S\}=\sum\limits_{T\subseteq S}(-1)^{|T|-1}\max\{T\}$$
$$\max_k\{S\}=\sum\limits_{T\subseteq S}(-1)^{|T|-k}\binom{|T|-1}{k-1}\min\{T\}$$
$$\min_k\{S\}=\sum\limits_{T\subseteq S}(-1)^{|T|-k}\binom{|T|-1}{k-1}\max\{T\}$$
$$f(S)=\sum\limits_{T\subseteq S}g(T)\Rightarrow g(S)=\sum\limits_{T\subseteq S}(-1)^{|S|-|T|}f(T)$$
$$f(S)=\sum\limits_{S\subseteq T}g(T)\Rightarrow g(S)=\sum\limits_{S\subseteq T}(-1)^{|S|-|T|}f(T)$$
设$\mu(S)=(-1)^{|S|}[S中没有重复元素]$
$$f(S)=\sum\limits_{T\subseteq S}g(T)\Rightarrow g(S)=\sum\limits_{T\subseteq S}\mu(S-T)f(T)$$
$$f(S)=\sum\limits_{S\subseteq T}g(T)\Rightarrow g(S)=\sum\limits_{S\subseteq T}\mu(T-S)f(T)$$
原文：https://www.cnblogs.com/JDFZ-ZZ/p/14191385.html