Combinatorics Through Guided Discovery, 2004a

Contents
About the Author
Preface
Preface to PreTeXt edition
v
vii
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1 What is Combinatorics? 1
1.1 About These Notes ............................. 1
1.2 Basic Counting Principles ......................... 2
1.2.1 The sum and product principles ................. 5
1.2.2 Functions and directed graphs .................. 9
1.2.3 The bijection principle ....................... 11
1.2.4 Counting subsets of a set ..................... 11
1.2.5 Pascal’s Triangle .......................... 12
1.2.6 The quotient principle ....................... 15
1.3 Some Applications of the Basic Principles ................ 20
1.3.1 Lattice paths and Catalan Numbers ............... 20
1.3.2 The Binomial Theorem ....................... 23
1.3.3 The pigeonhole principle ..................... 25
1.3.4 Ramsey Numbers .......................... 26
1.4 Supplementary Chapter Problems .................... 28
2 Applications of Induction and Recursion in Combinatorics and Graph
Theory 31
2.1 Some Examples of Mathematical Induction ............... 31
2.1.1 Mathematical induction ...................... 31
2.1.2 Binomial Coefficients and the Binomial Theorem ........ 33
2.1.3 Inductive definition ........................ 33
2.1.4 Proving the general product principle (Optional) ........ 34
2.1.5 Double Induction and Ramsey Numbers ............ 35
2.1.6 A bit of asymptotic combinatorics ................ 36
2.2 Recurrence Relations ............................ 37
2.2.1 Examples of recurrence relations ................. 38
2.2.2 Arithmetic Series (optional) .................... 39
2.2.3 First order linear recurrences ................... 40
2.2.4 Geometric Series .......................... 40
2.3 Graphs and Trees .............................. 41
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