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| Berman G., Fryer K.D. — Introduction to Combinatorics |
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| Предметный указатель |
Affine planes axioms 189 190
Affine planes coordinates 189
Affine planes exercises 189 193
Arithmetic power series discussion 209
Arithmetic power series exercises 211
Arithmetic power series summation method 209
Arrangements circular 39
Arrangements definition 38
Arrangements examples 40
Arrangements exercises 41
Arrangements number of r-arrangements 38
Arrangements with repetitions 39 40
Associative laws, finite fields 181
Averaging method examples 137
Averaging method existence 137
Balanced incomplete block designs blocks 265
Balanced incomplete block designs definition 265
Balanced incomplete block designs examples 267
Balanced incomplete block designs exercises 267
Balanced incomplete block designs Steiner triple system 267
Balanced incomplete block designs varieties 265
Binomial coefficients examples 52
Binomial coefficients exercises 47 54
Binomial coefficients Gaussian 218 221
Binomial coefficients identities 52 53
Binomial coefficients maximum term 51 52
Binomial coefficients Pascal's formula 48
Binomial distribution definition 213
Binomial distribution examples 211
Binomial distribution exercises 213
Binomial identities, exercises 55
Binomial random walk, example 235
Binomial theorem examples 46
Binomial theorem exercises 47
Binomial theorem formulation 45
Binomial theorem maximum term 45
Bipartite graph, definition 259
Birkhoff's theorem chromatic polynomial 18
Birkhoff's theorem proof 20
Birth and death process, examples 241
Blocks, balanced incomplete block designs 265
Boundaries, maps 18
Branches, trees 13
Cayley, counting labeled trees 15
Chessboard, grains of wheat caper 7
Chromatic polynomials Birkhoff's theorem 18
Chromatic polynomials complete graph 97
Chromatic polynomials definition 18
Chromatic polynomials empty graph 97
Chromatic polynomials examples 18 19 68 98 101—105 169
Chromatic polynomials exercises 21 69 106 173
Chromatic polynomials graphs 21 97 168
Chromatic polynomials inclusion-exclusion principle 68
Chromatic polynomials n-gon 170
Chromatic polynomials overlapping graphs 169
Chromatic polynomials properties 171
Chromatic polynomials recurrence relations 100 105
Chromatic polynomials trees 21 170
Chromatic polynomials unsolved problems 172
Chromatic triangles exercises 174
Chromatic triangles Ramsey problem 174
Coloring graphs 167
Coloring maps, exercises 21 167
Coloring planar maps 167
Combinations definition 42
Combinations examples 42 82 119 215
Combinations exercises 43 83 85 216 240
Combinations symbols 34
Combinations with repetitions 82
Combinations without repetitions 42
Commutative laws finite fields 181
Commutative laws Pappus' theorem 194
Complement, sets 60
Computer evaluating polynomials 24
Computer exercises 26 56 59 70 108 229 232 245 256 260 263 275
Conjectures, number of regions of plane 10
Continued fractions, exercises 86
Contradiction method examples 138
Contradiction method exercises 139
Contradiction method existence 138
Convex polyhedron, definition 155
Convex sets convex hull 140
Convex sets convex polygons 230
Convex sets definition 140
Convex sets exercises 142
Convex sets existence 140
Coordinates affine planes 189
Coordinates Desargues' theorem examples 191
Coordinates exercises 193
Coordinates homogeneous 193
Coordinates Pappus' theorem 194
Counting hairs Dirichlet's drawer principle 22
Counting hairs Fibonacci sequences 249
Counting hares discussion 249
Counting hares exercises 249
Counting hares Fibonacci sequence 249
Data consistency examples 206
Data consistency exercises 207
Derangements definition 70
Derangements examples 70
Derangements exercises 72
Derivatives, generating functions 121
Desargues' theorem exercises 195
Desargues' theorem finite fields 194
Desargues' theorem statement 193
Difference equations definition 92
Difference equations examples 92 123 239 240
Difference equations exercises 93 115 120 124 232 241
Difference equations Fibonacci sequence 94 242
Difference equations generating functions 113
Difference equations initial conditions 242
Difference equations nonempty subsets of set 2
Difference equations random walks 29
Difference equations subsets of set 3
Difference methods, operators 90
Difference operator definition 90
Difference operator evaluating polynomials 25
Difference operator examples 90
Difference operator exercises 26
Difference sets definition 268
Difference sets examples 268—270
Difference sets exercises 270
Differential equations examples 122
Differential equations generating functions 122
Dirichlet's drawer principle counting hairs 22
Dirichlet's drawer principle examples 22 136
Dirichlet's drawer principle exercises 23 138
Dirichlet's drawer principle formulation 136
Distribution of objects into boxes examples 214 215
Distribution of objects into boxes exercises 216
Distributive law, finite fields 181
Division ring, Desargues' theorem 194
Edges graphs 20
Edges maps 18
Edges trees 13
Equivalence classes discussion 152
Equivalence classes examples 152 153
Equivalence classes exercises 154
Euler's formula examples 156—159
Euler's formula exercises 160
Euler's formula method of averaging 137
Euler's formula polyhedra 156
Events, examples 200
examples see “specific entries”
exercises see “specific entries”
Exhaustion examples 134
Exhaustion exercises 135 185
Exhaustion nonexistence 134
Existence averaging method 137
Existence by construction 129
| Existence by exhaustion 133
Existence by trial and error 134
Existence contradiction method 138
Existence convex sets 144
Existence examples 129 133 134
Existence exercises 146 148 167
Existence maps 158 161 165
Existence orthogonal Latin squares 262
Existence projective planes 188 192
Existence regular polyhedra 163
Existence tessellations 147
Existence trees 22
Factorials, Stirling's formula 58
Fano plane examples 186
Fano plane finite 186
Fibonacci algorithm example 253
Fibonacci algorithm exercises 256
Fibonacci algorithm formulation 255
Fibonacci sequences counting hares 249
Fibonacci sequences difference equation 94
Fibonacci sequences examples 118 243 247
Fibonacci sequences exercises 95 120 245 246 249
Fibonacci sequences general term 95
Fibonacci sequences generating functions 243
Fibonacci sequences maximum and minimum 250
Fibonacci sequences representations 242
Fibonacci sequences sequences of plus and minus signs 247
Finite fields associate laws 181
Finite fields cardinality 184
Finite fields characteristic 139 184 195
Finite fields commutative laws 181
Finite fields definition 180 296
Finite fields discussion 180
Finite fields distributive law 181
Finite fields examples 131 138 182 184
Finite fields exercises 132 139 185
Finite planes exercises 189
Finite planes Fano plane 186
Formal power series, examples 3
Four-color problem discussion 164
Four-color problem trivalent maps 165
Gaussian binomial coefficients binomial coefficients 221
Gaussian binomial coefficients definition 218
Gaussian binomial coefficients examples 218—223
Gaussian binomial coefficients exercises 224
Generating functions combinatorial identities 116
Generating functions derivatives 121
Generating functions difference equations 113
Generating functions differential equations 122
Generating functions examples 3 109 111 113 114 116 119 121—124 215 227 228 234
Generating functions exercises 8 112 115 118 120 124 229
Generating functions exponential 110
Generating functions Fibonacci sequences 243
Generating functions random walks 29 233
Generating functions Taylor series 112
Generating functions triangulations 231
Genus, definition 165
Grains of wheat caper exercises 8
Grains of wheat caper induction 7
Grains of wheat caper story 7
Graphs bipartite 259
Graphs chromatic polynomials 21 97 168
Graphs coloring 167
Graphs edges 20
Graphs examples 169
Graphs exercises 168
Graphs labeled 20
Graphs overlapping 169
Graphs planar 167
Graphs vertices 20
Graphs wheel 168
Heawood, coloring theorem 166
Identities binomial coefficients 52
Identities combinatorial 116
Identities generating functions 116
Identity operator, definition 90
Inclusion-exclusion principle applications 67
Inclusion-exclusion principle calculus of sets 60
Inclusion-exclusion principle derangements 71
Inclusion-exclusion principle examples 64 66 83
Inclusion-exclusion principle exercises 66 69 81
Inclusion-exclusion principle linear equations 68 78
Inclusion-exclusion principle number theory 67
Inclusion-exclusion principle positive sets 64
Inclusion-exclusion principle probability 201
Inclusion-exclusion principle symmetric properties 79
Induction exercises 8
Induction number of nonempty subsets of set 2
Induction random walks 28
Induction regions of plane 10
Induction Tower of Hanoi puzzle 5
Initial conditions Fibonacci equation 242
Initial conditions random walks 29
Interval of uncertainty, definition 251
Irreducible, quadratic 183
Iteration examples 171
Iteration exercises 89
Iteration recurrence relations 87
Kirkman schoolgirl problem exercises 264
Kirkman schoolgirl problem formulation 263
Labeled graphs, example 20
Labeled maps, example 20
Labeled trees definition 14
Labeled trees enumeration 13
Labeled trees example 14
Labeled trees exercises 17
Labeling figures, exercises 135
Labeling Sperner's lemma 175
Latin squares definition 130 260
Latin squares examples 131 260
Latin squares exercises 132 263
Latin squares existence 262
Latin squares nonexistence 262
Latin squares orthogonal 131 261
Line at infinity, definition 192
Linear equations bounded solutions 79
Linear equations examples 74—79 215
Linear equations exercises 70 76 80 229
Linear equations inclusion-exclusion principle 68 78
Linear equations nonnegative solutions 74
Linear equations positive solutions 73
Linear equations solutions bounded below 73 76 78
Magic squares construction 272
Magic squares definition 271
Magic squares examples 271
Magic squares exercises 272
Maps boundaries 18
Maps coloring 21
Maps edges 18
Maps Euler's formula 156
Maps examples 129 157 158 159
Maps exercises 132 160 167
Maps existence 158 161
Maps five-color theorem 165
Maps four-color problem 164
Maps Heawood theorem 116
Maps Heawood theorem on sphere 21 155
Maps labeled 20
Maps on surface of genus 166
Maps ordinary 18
Maps planar 164
Maps polyhedral 157
Maps regular 161
Matchings exercises 260
Matchings maximal 259
Maximum and minimum examples 250 252
Maximum and minimum Fibonacci sequences 250
Mean position, random walks 29
Mercator projection, exercise 161
Monic polynomial, definition 27
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