莫比烏斯反演

\[[n=1] = \sum_{d|n}\mu(d) \]\[G(n) = \sum_{d|n}F(d) \Leftrightarrow F(n) = \sum_{d|n}\mu(\frac{n}{d})G(d) \]

二項式反演

\[G(n) = \sum_{i=0}^{n}\tbinom{n}{i}(-1)^{i}F(i) \Leftrightarrow F(n) = \sum_{i=0}^{n}\tbinom{n}{i}(-1)^{i}G(i) \]\[G(n) = \sum_{i=0}^{n}\tbinom{n}{i}F(i) \Leftrightarrow F(n) = \sum_{i=0}^{n}\tbinom{n}{i}(-1)^{n-i}G(i) \]\[G(n) = \sum_{i=n}\tbinom{i}{n}F(i) \Leftrightarrow F(n) = \sum_{i=n}\tbinom{i}{n}(-1)^{i-n}G(i) \]

Min-Max反演

\[\max(S) = \sum_{T \subseteq S} (-1)^{|T|+1}\min(T) \]\[\min(S) = \sum_{T \subseteq S} (-1)^{|T|+1}\max(T) \]

Kth反演

\[kthmax(S) = \sum_{T\subseteq S} (-1) ^{|T|-k}\tbinom{|T|-1}{k - 1}\min(T) \]\[kthmin(S) = \sum_{T\subseteq S} (-1) ^{|T|-k}\tbinom{|T|-1}{k - 1}\max(T) \]

斯特林反演

\[F(n) = \sum_{i=0}^{n} \begin{Bmatrix}n\\i\end{Bmatrix} G(i) \Leftrightarrow G(n) = \sum_{i=0}^{n} (-1)^{n-i}\begin{bmatrix}n\\i\end{bmatrix}F(i) \]\[F(n) = \sum_{i=0}^{n} (-1)^{n-i}\begin{Bmatrix}n\\i\end{Bmatrix} G(i) \Leftrightarrow G(n) = \sum_{i=0}^{n} \begin{bmatrix}n\\i\end{bmatrix}F(i) \]\[F(n) = \sum_{i=n}^{} \begin{Bmatrix}i\\n\end{Bmatrix} G(i) \Leftrightarrow G(n) = \sum_{i=n}^{} (-1)^{i-n}\begin{bmatrix}i\\n\end{bmatrix}F(i) \]\[F(n) = \sum_{i=n}^{}(-1)^{i-n} \begin{Bmatrix}i\\n\end{Bmatrix} G(i) \Leftrightarrow G(n) = \sum_{i=n}^{} \begin{bmatrix}i\\n\end{bmatrix}F(i) \]