数学公式合集

组合数

\[(a+b)^n=\sum_{i=0}^n\binom nia^{n-i}b^i \]

\[g(n)=\sum_{i=0}^n\binom nif(i)\Leftrightarrow f(n)=\sum_{i=0}^n(-1)^i\binom nig(i) \]

莫比乌斯反演

\[f(n)=\sum_{d|n}g(d)\Leftrightarrow g(n)=\sum_{d|n}\mu(d)f(\frac nd) \]

\[f(n)=\sum_{n|d}g(d)\Leftrightarrow g(n)=\sum_{n|d}\mu(\frac dn)f(d) \]

\[\sum_{d|n}\mu(d)=[n=1] \]

\[[\gcd(i,j)=1]=\sum_{d|\gcd(i,j)}\mu(d) \]

最值反演

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

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

\[\max(S)_k=\sum_{T\subseteq S,|T|\ge k}(-1)^{|T|-k}\binom{|T|-1}{k-1}\min(T) \]

\[\min(S)_k=\sum_{T\subseteq S,|T|\ge k}(-1)^{|T|-k}\binom{|T|-1}{k-1}\max(T) \]

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

\[\operatorname{lcm}(S)=\prod_{T\subseteq S}\gcd(T)^{(-1)^{|T|-1}} \]

斐波那契数列

\[若f(0)=0,f(1)=1,f(n)=af(n-1)+bf(n-2),\gcd(a,b)=1\\则\gcd(f(x),f(y))=f(\gcd(x,y)) \]

\[\sum_{i=0}^n\binom{n-i}i=F_{n+1} \]

下降幂和斯特林数

\[\binom nkk^{\underline i}=\binom{n-i}{k-i}n^{\underline i} \]

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

\[x^n=\sum_{k=0}^n\begin{Bmatrix}n\\k\end{Bmatrix}x^{\underline k} \]

\[\begin{bmatrix}n\\m\end{bmatrix}=\begin{bmatrix}n-1\\m-1\end{bmatrix}+(n-1)\begin{bmatrix}n-1\\m\end{bmatrix} \]

\[x^{\underline n}=\sum_{k=0}^n\begin{bmatrix}n\\k\end{bmatrix}(-1)^{n-k}x^k \]

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

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

狄利克雷卷积

\[\mu*1=\epsilon,\varphi*1=\operatorname{id} \]

\[S(n)=\sum_{i=1}^nf(i),g(1)S(n)=\sum_{i=1}^n(f*g)(i)-\sum_{i=2}^ng(i)S(\lfloor\frac ni\rfloor) \]