The Art of Counting Bijections, Double Counting
Peng Shi
Department of Mathematics Duke University
September 16, 2009 Straightforward, careful counting Bijection Counting in multiple ways
Today, we will consider some commonly-used paradigms of counting:
What we focus on in this talk?
Enumerative combinatorics is a huge branch of mathematics, involving many theorems and techniques, which we cannot hope to cover in one class
Peng Shi, Duke University The Art of Counting, Bijections, Double Counting 2/13 Bijection Counting in multiple ways
What we focus on in this talk?
Enumerative combinatorics is a huge branch of mathematics, involving many theorems and techniques, which we cannot hope to cover in one class
Today, we will consider some commonly-used paradigms of counting: Straightforward, careful counting
Peng Shi, Duke University The Art of Counting, Bijections, Double Counting 2/13 Counting in multiple ways
What we focus on in this talk?
Enumerative combinatorics is a huge branch of mathematics, involving many theorems and techniques, which we cannot hope to cover in one class
Today, we will consider some commonly-used paradigms of counting: Straightforward, careful counting Bijection
Peng Shi, Duke University The Art of Counting, Bijections, Double Counting 2/13 What we focus on in this talk?
Enumerative combinatorics is a huge branch of mathematics, involving many theorems and techniques, which we cannot hope to cover in one class
Today, we will consider some commonly-used paradigms of counting: Straightforward, careful counting Bijection Counting in multiple ways
Peng Shi, Duke University The Art of Counting, Bijections, Double Counting 2/13 Sum Rule: S S S T If A = A1 A2 ··· An, Ai Aj = ∅, then
|A| = |A1| + |A2| + ··· + |An|
Product Rule: If W = W1 × W2 × · · · × Wn (Cartesian set product), then
|W | = |W1||W2| · · · |Wn|
Paradigm 1: Careful Straightforward Counting
Comprehensive enumeration/case work Make sure to count every case Don’t double count
Peng Shi, Duke University The Art of Counting, Bijections, Double Counting 3/13 Product Rule: If W = W1 × W2 × · · · × Wn (Cartesian set product), then
|W | = |W1||W2| · · · |Wn|
Paradigm 1: Careful Straightforward Counting
Comprehensive enumeration/case work Make sure to count every case Don’t double count
Sum Rule: S S S T If A = A1 A2 ··· An, Ai Aj = ∅, then
|A| = |A1| + |A2| + ··· + |An|
Peng Shi, Duke University The Art of Counting, Bijections, Double Counting 3/13 Paradigm 1: Careful Straightforward Counting
Comprehensive enumeration/case work Make sure to count every case Don’t double count
Sum Rule: S S S T If A = A1 A2 ··· An, Ai Aj = ∅, then
|A| = |A1| + |A2| + ··· + |An|
Product Rule: If W = W1 × W2 × · · · × Wn (Cartesian set product), then
|W | = |W1||W2| · · · |Wn|
Peng Shi, Duke University The Art of Counting, Bijections, Double Counting 3/13 n ways to choose 1st element, n − 1 ways to choose 2nd, . . . So n(n − 1) ··· (n − m + 1)? But each subset is counted m! times