Min-Max 容斥可以将 Min 和 Max 互相转换。

max⁡i∈Sxi=∑T⊆S(−1)∣T∣−1min⁡j∈Txj\max_{i\in S}{x_i}=\sum_{T\subseteq S}{(-1)^{|T|-1}\min_{j\in T}{x_j}} min⁡i∈Sxi=∑T⊆S(−1)∣T∣−1max⁡j∈Txj\min_{i\in S}{x_i}=\sum_{T\subseteq S}{(-1)^{|T|-1}\max_{j\in T}{x_j}}

应用的话就是期望的时候可以套用：

E(max⁡i∈Sxi)=∑T⊆S(−1)∣T∣−1E(min⁡j∈Txj)E\left(\max_{i\in S}{x_i}\right)=\sum_{T\subseteq S}{(-1)^{|T|-1}E\left(\min_{j\in T}{x_j} \right)} E(min⁡i∈Sxi)=∑T⊆S(−1)∣T∣−1E(max⁡j∈Txj)E\left(\min_{i\in S}{x_i}\right)=\sum_{T\subseteq S}{(-1)^{|T|-1}E\left(\max_{j\in T}{x_j} \right)}

还有一种形式：

kthmax⁡xii∈S=∑T⊆S(−1)∣T∣−k(∣T∣−1k−1)min⁡j∈Txj\underset{i\in S}{\operatorname{kthmax}{x_i}}=\sum_{T\subseteq S}{(-1)^{|T|-k}\dbinom {|T|-1}{k-1}\min_{j\in T}{x_j}} kthmin⁡xii∈S=∑T⊆S(−1)∣T∣−k(∣T∣−1k−1)max⁡j∈Txj\underset{i\in S}{\operatorname{kthmin}{x_i}}=\sum_{T\subseteq S}{(-1)^{|T|-k}\dbinom {|T|-1}{k-1}\max_{j\in T}{x_j}} E(kthmax⁡xii∈S)=∑T⊆S(−1)∣T∣−k(∣T∣−1k−1)E(min⁡j∈Txj)E\left(\underset{i\in S}{\operatorname{kthmax}{x_i}}\right)=\sum_{T\subseteq S}{(-1)^{|T|-k}\dbinom {|T|-1}{k-1}E\left(\min_{j\in T}{x_j}\right)} E(kthmin⁡xii∈S)=∑T⊆S(−1)∣T∣−k(∣T∣−1k−1)E(max⁡j∈Txj)E\left(\underset{i\in S}{\operatorname{kthmin}{x_i}}\right)=\sum_{T\subseteq S}{(-1)^{|T|-k}\dbinom {|T|-1}{k-1}E\left(\max_{j\in T}{x_j}\right)}