南京大学：《组合数学》课程教学资源（课堂讲义）筛法 Sieve methods

The Sieve Methods

The Sieve Methods

PIE (Principle of Inclusion-Exclusion) |AUB=A+|B|-A∩B AUBUC=A+B+C -A∩B-A∩C-B∩C +AnBnC C BnC AnC AnBnC B AnB

PIE (Principle of Inclusion-Exclusion) |A ￾ B| = |A| + |B| ￾|A ⇥ B| |A ￾ B ￾ C| = |A| + |B| + |C| ￾|A ⇥ B| ￾ |A ⇥ C| ￾ |B ⇥ C| +|A ￾ B ￾ C|

PIE (Principle of Inclusion-Exclusion) n= ∑|A-∑IA,nA+ 二1 1≤i≤m 1≤i<j≤n …+(-1)n-1|A1∩…nAnl =∑(--1∩A: IC{1,…,n} I≠0

PIE (Principle of Inclusion-Exclusion) ￾ ￾ ￾ ￾ ￾ ⇥ n i=1 Ai ￾ ￾ ￾ ￾ ￾ = ￾ 1￾i￾n |Ai| ￾ ￾ 1￾i<j￾n |Ai ⇥ Aj |+ ··· + (￾1)n￾1|A1 ⇤ ··· ⇤ An| = ￾ I￾{1,...,n} I￾=￾ (￾1)|I|￾1 ￾ ￾ ￾ ￾ ￾ ￾ i￾I Ai ￾ ￾ ￾ ￾ ￾

PIE (Principle of Inclusion-Exclusion) A1,A2,...,An CU universe 元nn周-U4 -2,-y-1QA I{1，，n} I≠0 A虹=∩A, Ao=U i∈I

PIE (Principle of Inclusion-Exclusion) A1, A2,...,An ￾ U universe ￾ ￾A1 ⇥ A2 ⇥ ··· An ￾ ￾ = ￾ ￾ ￾ ￾ ￾ U ￾ ⇥ n i=1 Ai ￾ ￾ ￾ ￾ ￾ AI = ￾ i￾I Ai A￾ = U = |U| ￾ ￾ I￾{1,...,n} I￾=￾ (￾1)|I|￾1 ￾ ￾ ￾ ￾ ￾ ￾ i￾I Ai ￾ ￾ ￾ ￾ ￾

PIE (Principle of Inclusion-Exclusion) A1,A2,...,An U universe AinA2n…An=(-1)川|Al IC{1,.,n} Ar=∩A, Ao=U i∈1

PIE (Principle of Inclusion-Exclusion) A1, A2,...,An ￾ U universe ￾ ￾A1 ⇥ A2 ⇥ ··· An ￾ ￾ = AI = ￾ i￾I Ai A￾ = U ￾ I￾{1,...,n} (￾1)|I| |AI |

PIE (Principle of Inclusion-Exclusion) A1,A2,:.,AnCU←—universe AinA2∩…An=S0-S1+S2+·+(-1)rSn A=∩A: Ao=U i∈I Sk=∑IA So |Aol Ul I=k

PIE (Principle of Inclusion-Exclusion) A1, A2,...,An ￾ U universe ￾ ￾A1 ⇥ A2 ⇥ ··· An ￾ ￾ = AI = ￾ i￾I Ai A￾ = U Sk = ￾ |I|=k |AI | S0 = |A￾| = |U| S0 ￾ S1 + S2 + ··· + (￾1)nSn

Surjections #of f:[n]onto, m] U=[nl→[ml A,=[m]→([m{) 0- ∑(-1)1A iElm] IE[m] Ar=∩A, Ao=U i∈I

Surjections f : [n] onto ￾￾￾⇥ [m] # of U = [n] ￾ [m] Ai = [n] ￾ ([m] \ {i}) AI = ￾ i￾I Ai A￾ = U ￾ I￾[m] (￾1)|I| |AI | ￾ ￾ ￾ ￾ ￾ ￾ ￾ i￾[m] Ai ￾ ￾ ￾ ￾ ￾ ￾ =

Surjections U=[nl→[ml A:=[n]→(ml\{i}) Ag=UAr=∩A:=[m→([m\I) i∈I |Ar=(m-1I)' ∩， ∑(-1)4 i∈[ml IC[m] =∑-0(m-”--()m- IC[m] 三-*()

Surjections U = [n] ￾ [m] Ai = [n] ￾ ([m] \ {i}) AI = ￾ i￾I A￾ = U Ai = [n] ￾ ([m] \ I) |AI | = (m ￾ |I|) n = ⇤ m k=1 (￾1)m￾k ￾m k ⇥ kn = ￾ I￾[m] (￾1)|I| (m ￾ |I|) n ￾ I￾[m] (￾1)|I| |AI | ￾ ￾ ￾ ￾ ￾ ￾ ￾ i￾[m] Ai ￾ ￾ ￾ ￾ ￾ ￾ = = ￾ m k=0 (￾1)k ￾m k ￾ (m ￾ k) n

Surjections m (f-1(0),f-1(1),.,f-1(m-1) ordered partition of [m] 网s例=m{n} {}-点2-1-()

Surjections = ⇤ m k=1 (￾1)m￾k ￾m k ⇥ kn ￾ ￾ ￾[n] onto ￾￾￾⇥ [m] ￾ ￾ ￾ (f ￾1(0), f ￾1(1),...,f ￾1(m ￾ 1)) ordered partition of [m] ￾ ￾ ￾[n] onto ￾￾￾￾ [m] ￾ ￾ ￾ = m! ￾ n m ￾ ￾ n m ￾ = 1 m! ￾ m k=1 (￾1)m￾k ￾m k ￾ kn

Derangement les problemes des rencontres ESSAY DANALYSE Two decks,A and B,of cards: SUR LES JEUX DE HAZARD. The cards of A are laid out in a row, Par M.Remond de Montmort. and those of B are placed at random, one at the top on each card of A. What is the probability that no 2 cards are the same in each pair?

Derangement Two decks, A and B, of cards: The cards of A are laid out in a row, and those of B are placed at random, one at the top on each card of A. What is the probability that no 2 cards are the same in each pair? les problèmes des rencontrés :