Catalan
Stirling
容斥
- UVA10325 The Lottery(状压+容斥)
- UVA11806 Cheerleaders
- SP4191 MSKYCODE - Sky Code
- [CQOI2015]选数(容斥+递推)
- [SCOI2010]幸运数字
莫比乌斯反演与筛法
\[g(n)=\sum_{d|n}f(d)\] \[f(n)=\sum_{d|n}{\mu(d)g(\frac{n}{d})}\]
blogs:
题目:
- HDU1695 GCD + [HAOI2011]Problem b + [POI2007]ZAP-Queries
三倍经验 - [SDOI2015]约数个数和
- Crash的数字表格
- P2257 YY的GCD
- 莫比乌斯函数之和
- 4805: 欧拉函数求和
- SP4168 SQFREE - Square-free integers
- 2440: [中山市选2011]完全平方数
- [SDOI2014]数表(莫比乌斯反演+树状数组)
- P3768 简单的数学题
- 3309: DZY Loves Math
- P4240 毒瘤之神的考验(真的毒瘤的预处理啊orz)
- [NOI2016]循环之美
- [SDOI2017]数字表格(关于幂的莫比乌斯反演orz)
二项式反演
\[f(n)=\sum_{k=p}^n (\begin{matrix} n \\ k \end{matrix})g(k)\]
\[g(n)=\sum_{k=p}^n(-1)^{n-k}(\begin{matrix} n \\ k \end{matrix})f(k)\]
莫比乌斯函数、二项式、斯特林数以及它们的反演
公式:
- \[g(n)=\sum_{d|n}f(d)\] \[f(n)=\sum_{d|n}{\mu(d)g(\frac{n}{d})}\]
- \[f(n)=\sum_{k=p}^n (\begin{matrix} n \\ k \end{matrix})g(k)\]
\[g(n)=\sum_{k=p}^n(-1)^{n-k}(\begin{matrix} n \\ k \end{matrix})f(k)\] - \[g(n)=\sum_{n|d}{f(d)[d \leq m]}\] \[f(n)=\sum_{n|d}{\mu(\frac{d}{n})g(d)[d \leq m]}\]
- 如果\(f(n)\)是积性函数,且\((x, y) = 1\),则有
\[f(xy)=f(x)f(y)\] - \[\sum_{i=1}^{n}{i[(i, n) == 1]}= \frac{\varphi(n)*n}2\]
(用到结论:\(if (i, n) == 1, then (n-i, n) = 1\)) - \[d(ij)=\sum_{x|n}{\sum_{y|n} [(x, y) == 1]}\]
- \[\sum_{i=1}^{n}{i \times \lfloor \frac{n}{i} \rfloor} = \sum_{i=1}^n{\frac{\lfloor \frac{n}{i} \rfloor(\lfloor \frac{n}{i} \rfloor + 1)}{2}}\]
- \[(id*\mu)(i)=\varphi(i)\]
\[(\varphi*I)(i)=id(i)\]
\[(\mu*I)(i)=e(i)\] - \[[n == 1]=\sum_{d|n}{\mu(d)}\]
- \[n=\sum_{d|n}{\varphi(d)}\]
- \[\sum_{i=1}^{n}{\sum_{j=1}^{m}ij}=\frac{n^2(n+1)^2}{4}\]
- 除数函数 \[\sigma_k(n)=\sum_{d|n}{d^k}\]
约数个数函数 \[\tau(n)=\sigma_0(n)=\sum_{d|n}1\]
约数和函数\[\sigma(n)=\sigma_1(n)=\sum_{d|n}d\] - \[\sum_{i=1}^ni=\frac{n(n+1)}{2}\]
\[\sum_{i=1}^ni^2=\frac{(n+1)(2n+1)n}{6}\]
\[\sum_{i=1}^ni^3=\frac{n^2(n+1)^2}{4}\] - \[\varphi(ij)=\frac{\varphi(i)\varphi(j)(i,j)}{\varphi((i,j))}\]
- \[[f(x) == 1] = e(f(x))=(\mu*I)(f(x))\](常用于引进\(\mu\)以进行莫比乌斯反演,如[NOI2016]循环之美)