Combinatorics Through Guided Discovery, 2004a

Contents
xv
5 The Principle of Inclusion and Exclusion 93
5.1 The size of a union of sets ......................... 93
5.1.1 Unions of two or three sets .................... 93
5.1.2 Unions of an arbitrary number of sets .............. 94
5.1.3 The Principle of Inclusion and Exclusion ............ 95
5.2 Application of Inclusion and Exclusion ................. 97
5.2.1 Multisets with restricted numbers of elements ......... 97
5.2.2 The Menage Problem ........................ 97
5.2.3 Counting onto functions ...................... 97
5.2.4 The chromatic polynomial of a graph .............. 98
5.3 Deletion-Contraction and the Chromatic Polynomial .......... 99
5.4 Supplementary Problems .........................101
6 Groups acting on sets 103
6.1 Permutation Groups ............................103
6.1.1 The rotations of a square ......................103
6.1.2 Groups of Permutations ......................104
6.1.3 The symmetric group .......................106
6.1.4 The dihedral group .........................107
6.1.5 Group tables (Optional) ......................110
6.1.6 Subgroups ..............................111
6.1.7 The cycle structure of a permutation ...............112
6.2 Groups Acting on Sets ...........................114
6.2.1 Groups acting on colorings of sets ................116
6.2.2 Orbits ................................119
6.2.3 The Cauchy-Frobenius-Burnside Theorem ...........122
6.3 Pólya-Redfield Enumeration Theory ...................125
6.3.1 The Orbit-Fixed Point Theorem ..................126
6.3.2 The Pólya-Redfield Theorem ...................128
6.4 Supplementary Problems .........................131
A Relations 133
A.1 Relations as sets of Ordered Pairs .....................133
A.1.1 The relation of a function .....................133
A.1.2 Directed graphs ...........................135
A.1.3 Digraphs of Functions .......................136
A.2 Equivalence relations ............................138
B Mathematical Induction 145
B.1 The Principle of Mathematical Induction .................145
B.1.1 The ideas behind mathematical induction ............145
B.1.2 Mathematical induction ......................147
B.1.3 Proving algebraic statements by induction ...........148
B.1.4 Strong Induction ..........................149