Combinatorics Through Guided Discovery, 2004a

xiv
Contents
2.3.1 Undirected graphs ......................... 41
2.3.2 Walks and paths in graphs .................... 43
2.3.3 Counting vertices, edges, and paths in trees ........... 43
2.3.4 Spanning trees ........................... 45
2.3.5 Minimum cost spanning trees ................... 46
2.3.6 The deletion/contraction recurrence for spanning trees .... 47
2.3.7 Shortest paths in graphs ...................... 48
2.4 Supplementary Problems ......................... 49
3 Distribution Problems 51
3.1 The idea of a distribution .......................... 51
3.1.1 The twentyfold way ........................ 51
3.1.2 Ordered functions ......................... 54
3.1.3 Multisets ............................... 56
3.1.4 Compositions of integers ..................... 56
3.1.5 Broken permutations and Lah numbers ............. 57
3.2 Partitions and Stirling Numbers ...................... 58
3.2.1 Stirling Numbers of the second kind ............... 58
3.2.2 Stirling Numbers and onto functions ............... 60
3.2.3 Stirling Numbers and bases for polynomials .......... 61
3.3 Partitions of Integers ............................ 63
3.3.1 The number of partitions of k into n parts ............ 63
3.3.2 Representations of partitions ................... 63
3.3.3 Ferrers and Young Diagrams and the conjugate of a partition 64
3.3.4 Partitions into distinct parts .................... 69
3.4 Supplementary Problems ......................... 70
4 Generating Functions 73
4.1 The Idea of Generating Functions ..................... 73
4.1.1 Visualizing Counting with Pictures ............... 73
4.1.2 Picture functions .......................... 74
4.1.3 Generating functions ........................ 75
4.1.4 Power series ............................. 77
4.1.5 Product principle for generating functions ........... 78
4.1.6 The extended binomial theorem and multisets ......... 79
4.2 Generating functions for integer partitions ................ 81
4.3 Generating Functions and Recurrence Relations ............ 85
4.3.1 How generating functions are relevant .............. 85
4.3.2 Fibonacci numbers ......................... 86
4.3.3 Second order linear recurrence relations ............. 86
4.3.4 Partial fractions ........................... 87
4.3.5 Catalan Numbers .......................... 89
4.4 Supplementary Problems ......................... 90