Combinatorics Through Guided Discovery, 2004a

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Index 201 geometric progression, 40 geometric series, 40, 79 graph, 41 chromatic polynomial of, 99 coloring of, 98 proper, 98 complete, 26 connected component of, 98, 165 coordinate, 135 directed, 9, 135 distance in, 48 simple, 165 graphs isomorphic, 131 Gray Code, 28 greedy method, 46 group acting on a set, 115 group action on colorings, 118 group of permutations, 105 hatcheck problem, 95 homogeneous linear recurrence, 40 identity function, 104, 137 identity property, 105 identity property (for permutations), 105 inclusion and exclusion principle, 93 for unions of sets, 95 indicator polynomials, 151 induction double, 35 mathematical, the principle of, 31, 147 mathematical, the strong principle of, 32 strong double, 35 inductive conclusion, 32 hypothesis, 32 step, 32 inductive definition, 33 injection, 4, 134 inverse function, 105 inverse property, 105 involution, 114 isometry, 108 isomorphic graphs, 131 k-set structures, 160 Lah number, 57 lattice path, 20 diagonal, 20 length (of a path), 48 linear recurrence, 40, 87 constant coefficient, 40 homogeneous, 40 second order, 86 mathematical induction double, 35 principle of, 31, 147 strong double, 35 ménage problem, 97 method probabilistic, 36 minimum cost spanning tree, 46 monochromatic subgraph, 37 multinomial coefficient, 60 multiorbit, 120 multiorbits, 127 multiplicity in a multiset, 56 multiset, 56 multisets product principle, 122 quotient principle, 122 sum principle, 121 union, 121 one-to-one, 4 one-to-one function, 134 onto function, 10, 134 counting, 60 ordered, 55 onto functions number of, 97 orbit, 119 orbit enumerator, 126 Orbit-Fixed Point Theorem, 127 ordered degree sequence, 72 ordered function, 55 ordered onto function, 55 ordered pair, 3 ordinary generating function, 152
202 Index pair structure, 159 pair,ordered, 3 partial fractions method of, 88 partition blocks of, 6 of a set, 5, 58 Stirling Numbers, 58 partition (of a set), 141 partition of a set type vector, 59 partition of an integer, 63 conjugate of, 65 decreasing list, 64 Ferrers diagram, 64 into n parts, 63 self conjugate, 65 type vector, 64 Young diagram, 64 partitions of a set number of, 59 Pascal’s Triangle, 12 path in graph, 43 lattice, 20 diagonal, 20 length of, 48 permutation k-element, 8 as a bĳection, 11 broken, 57 cycle of, 112 two row notation, 107 permutation group, 105 picture enumerator, 74 picture enumerators product principle for, 75 pigeonhole principle, 25 generalized, 26 Pólya-Redfield Theorem, 129 principle bĳection, 11 product, 5, 6 general, 6 quotient, 143 sum, 5, 6 principle of inclusion and exclusion, 93 for unions of sets, 95 principle of mathematical induction, 31, 147 probabilistic method, 36 product Cartesian, 5 product notation, 8 product principle, 5, 6 for multisets, 122 general, 6, 34 picture enumerators, 75 product principle for exponential generating functions, 160 product principle for generating functions, 79 progression arithmetic, 39 geometric, 40 proper coloring of a graph, 98 quotient principle, 18, 143 for multisets, 122 range (of a function), 133 recurrence, 38 constant coefficient, 86, 87 deletion-contraction, 48 linear, 40, 86, 87 linear homogeneous, 40 second order, 86, 87 solution to, 38 two variable, 58 recurrence relation, 38 recursive definition, 33 reflexive, 139 relation, 133 equivalence, 140, 142 of a function, 133 recurrence, 38 reflexive, 139, 140 transitive, 140 rising factorial power, 55 rotation group, 105 second order recurrence, 86 self-conjugate partition, 65
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Combinatorics Through Guided Discov
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Edition: First PreTeXt Edition Webs
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vi a great deal of time to helping
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viii because they use an important
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x you happen to get a hard problem
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xii Discovery had been placed on th
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xiv Contents 2.3.1 Undirected graph
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xvi Contents C Exponential Generati
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2 1. What is Combinatorics? learned
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4 1. What is Combinatorics? For exa
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6 1. What is Combinatorics? and the
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8 1. What is Combinatorics? + Probl
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10 1. What is Combinatorics? If we
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12 1. What is Combinatorics? are go
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14 1. What is Combinatorics? k =0 1
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16 1. What is Combinatorics? • Pr
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18 1. What is Combinatorics? • Pr
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20 1. What is Combinatorics? a Prov
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22 1. What is Combinatorics? (c) Fi
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24 1. What is Combinatorics? (vii)
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26 1. What is Combinatorics? · Pro
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28 1. What is Combinatorics? 1.4 Su
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30 1. What is Combinatorics?
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32 2. Applications of Induction and
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52 3. Distribution Problems problem
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54 3. Distribution Problems bĳecti
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56 3. Distribution Problems 3.1.3 M
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58 3. Distribution Problems broken
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60 3. Distribution Problems 3.2.2 S
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62 3. Distribution Problems coeffic
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64 3. Distribution Problems Problem
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66 3. Distribution Problems Problem
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68 3. Distribution Problems number
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70 3. Distribution Problems ⇒ Pro
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72 3. Distribution Problems ⇒ 12.
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74 4. Generating Functions • Prob
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78 4. Generating Functions x k in t
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84 4. Generating Functions the gene
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86 4. Generating Functions ◦ Prob
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88 4. Generating Functions Writing
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90 4. Generating Functions 4.4 Supp
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92 4. Generating Functions
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94 5. The Principle of Inclusion an
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100 5. The Principle of Inclusion a
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102 5. The Principle of Inclusion a
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104 6. Groups acting on sets on. We
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106 6. Groups acting on sets still
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108 6. Groups acting on sets Let us
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110 6. Groups acting on sets Proble
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112 6. Groups acting on sets 6.1.7
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114 6. Groups acting on sets ⇒ ·
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116 6. Groups acting on sets • Pr
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118 6. Groups acting on sets 3, and
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122 6. Groups acting on sets Taking
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126 6. Groups acting on sets of dif
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130 6. Groups acting on sets ⇒ Pr
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132 6. Groups acting on sets (c) Wr
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134 A. Relations Problem 329. Here
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136 A. Relations Problem 333. Draw
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138 A. Relations Problem 340. Expla
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140 A. Relations • If R is a rela
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142 A. Relations Problem 355. Suppo
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144 A. Relations
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146 B. Mathematical Induction (e) P
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148 B. Mathematical Induction shows
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