二项式反演

P1595信封问题

\[\begin{aligned} n!=\sum\limits_{i=0}^{n}\binom{n}{i}f(i)\\ f(i)=\sum\limits_{i=0}^n(-1)^{n-i}\binom{n}{i}i! \end{aligned} \]

P10596集合计数

\[\begin{aligned} \binom{N}{n}(2^{2^{N-n}}-1)=\sum\limits_{i=n}^N\binom{i}{n}f(i)\\ f(n)=\sum\limits_{i=n}^{N}(-1)^{i-n}\binom{i}{n}\binom{N}{i}(2^{2^{N-i}}-1) \end{aligned} \]

P5505分特产

\[\begin{aligned} \binom{N}{n}\prod\limits_{i=1}^M\binom{a_i+N-1-i}{N-i-1}=\sum\limits_{i=n}^{N}\binom{i}{n}f(i)\\ f(n)=\sum\limits_{i=n}^{N}(-1)^{i-n}\binom{i}{n}\binom{N}{i}\prod\limits_{j=1}^M\binom{a_j+N-1-j}{N-j-1} \end{aligned} \]

P4859已经没有什么好害怕的了

\[\begin{aligned} g(N, n)\times(N-n)!=\sum\limits_{i=n}^N\binom{i}{n}f(i)\\ f(n)=\sum\limits_{i=n}^N(-1)^{i-n}\binom{i}{n}g(N, i)\times(N-i)!\\ g(n, m)=g(n-1, m)+(sml_n-(m-1))g(n-1,m-1) \end{aligned} \]