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Charalambides C.A. — Enumerative Combinatorics
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Название: Enumerative Combinatorics
Автор: Charalambides C.A.
Аннотация: Enumerative Combinatorics presents elaborate and systematic coverage of the theory of enumeration. The first seven chapters provide the necessary background, including basic counting principles and techniques, elementary enumerative topics, and an extended presentation of generating functions and recurrence relations. The remaining seven chapters focus on more advanced topics, including, Stirling numbers, partitions of integers, partition polynomials, Eulerian numbers and Polya's counting theorem.
Extensively classroom tested, this text was designed for introductory- and intermediate-level courses in enumerative combinatorics, but the far-reaching applications of the subject also make the book useful to those in operational research, the physical and social science, and anyone who uses combinatorial methods. Remarks, discussions, tables, and numerous examples support the text, and a wealth of exercises-with hints and answers provided in an appendix—further illustrate the subject's concepts, theorems, and applications.
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Рубрика: Математика /
Статус предметного указателя: Готов указатель с номерами страниц
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Год издания: 2002
Количество страниц: 609
Добавлена в каталог: 18.11.2009
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Предметный указатель
Abel polynomials 206
Abel’s formula 207
Addition principle 16
Arithmetic progression, sum of terms of 31 194
Ballot numbers 81 230 274
Ballot problem 80 85 100
Banach matchbox problem 85
Bell numbers 97 199 342 474 477
Bell numbers, exponential 199
Beneficial bet 82
Bernoulli numbers 328 329
Binary number system 19 54
Binary sequences, enumeration of 19 36 54
Binomial coefficient (s) 110
Binomial coefficient (s), cycles of 231
Binomial coefficient (s), definition of 112
Binomial coefficient (s), monotonicity of 115
Binomial coefficient (s), orthogonality relation 126
Binomial coefficient (s), recurrence relation of 55 56 59 129
Binomial coefficient (s), table of 56
Binomial convolution formula 115
Binomial generating function 216
Binomial moments of a sequence 215
Bipartitional polynomials, exponential 454
Bipartitional polynomials, general 455
Bipartitional polynomials, logarithmic 457
Bipartitional polynomials, partial 454
Bipartitional polynomials, potential 458
Birthday problem 88
Bonferroni inequalities 154
Booking tickets 20
Boole inequality 155
Boolean algebra 12
Burnside theorem 494
Caravan in the desert 180
Cardinal of a set 14
Carlitz numbers 537
Carlitz numbers, definition of 523
Carlitz numbers, explicit expression of 523
Carlitz numbers, expressed by the generalized factorial coefficients 528
Carlitz numbers, generating function of 527
Carlitz numbers, limiting expression of 529
Carlitz numbers, non-central 539
Carlitz numbers, recurrence relation of 524
Carlitz numbers, table of 526
Carlitz polynomial 525
Carlitz-Riordan numbers of the first kind 329
Carlitz-Riordan numbers of the second kind 330 543 544
Cartesian product 5
Cashier problem 81
Catalan numbers 81 100 196 229
Catalan numbers, convolution of 229
Catalan numbers, generating function of 197
Catalan numbers, recurrence relation of 196
Catalan numbers, table of 196
Cauchy identity 465 481
Cauchy numbers 327 329
Cauchy’s binomial convolution formula 115
Cayley identity 481
Characteristic equation 239
Chebyshev polynomials of the first kind 449
Chebyshev polynomials of the second kind 450
Circular permutation(s) 92 462
Circular permutation(s), enumeration by successions 181 183
Circular permutation(s), succession of 181
Coefficients of the generalized factorials 345
Coefficients of the generalized factorials, associated 367
Coincidence numbers 173
Coincidence numbers, generating function of 219 222
Coincidence numbers, table of 173
Coincidences, double 186
Coincidences, generalized problem 188
Coincidences, multiple 187
Coincidences, number of 173
Coincidences, problem 173
Coloring(s) of a cube 496 506
Coloring(s) of a roulette 510
Coloring(s) of enumeration of 500
Coloring(s) of model of 499
Coloring(s) of the elements of a set 499
Coloring(s) of weight of 501
Combination (s) with limited repetition generating function for the number of 211
Combination (s) with repetition definition of 51
Combination (s) with restricted repetition enumeration of 139 144 164 504
Combination (s), definition of 51
Combination (s), enumeration of 58
Combination (s), general generating function for the number of 209
Combination (s), generating function for the number of 210
Combination (s), recurrence relation for the number of 59
Compositions of integers 401 511
Compound bivariate discrete distributions 456
Compound discrete distributions 422
Connection of cumulants and moments 426 457
Connection of ordinary and factorial moments 285
Convolution of a logarithmic distribution 287
Convolution of sequences 194
Correspondence 7
Coupon collector’s problem 151 313
Cycles of binomial coefficients 231
De Morgan’s formulae 12
Decomposition of a product of primes into factors 347
Delannoy numbers 274
Derangement(s), definition of 170
Derangement(s), number of 170
Derangement(s), recurrence relation for the number of 171
Derivatives of a composite function 420 453
Difference equation 22
Dinner choices in a Chinese restaurant 418
Display of flags on poles 60
Distribution of shares 83 262
Distributions of balls of general specification into distinguish, ordered urns enumeration of 348 349
Distributions of balls of general specification into distinguishable urns enumeration of 350—352
Distributions of balls of generating function of the number of 356
Distributions of distinguishable balls into distinguishable urns enumeration of 48 340 341
Distributions of indistinguish balls into distinguishable urns 60
Distributions of students in sections 67
Distributions, connection with integer solutions of a linear equation 71
Distributions, enumeration of 54 343— 345
Distributions, generating function of the number of 355
Division (s) of a finite set, definition of 13
Division (s) of a finite set, enumeration of 62 97 124 149
Division (s) of a finite set, recurrence relation for the number of 64
Double coincidences 186
Down-up permutations 543
Duration of a series of drawings 175
Elementary symmetric functions 192 427 432 458
Elements in a given number of exchangeable sets, enumeration of 143 147 148
Elements in a given number of sets, enumeration of 137 145 148
Elements in a union of exchangeable sets, enumeration of 142
Elements in a union of sets, enumeration of 132 134
Elements of a given rank 156
Elements of a given rank, enumeration of 156 157
Elevator passengers’ discharge 67 91 133
Equivalence class 493
Equivalence relation 7
Equivalent sets 9
Euler numbers 514 543
Eulerian numbers 530
Eulerian numbers, definition of 514
Eulerian polynomial 516
Euler’s function 140 491 497 498 506 510 511
Euler’s function, table of 140
Euler’s identity 393 402
Euler’s pentagonal theorem 389
Euler’s recurrence relation for sub-factorials 171
Exchangeable sets 142
Explicit expression of 515
Expressed by non-central Stirling numbers of the second kind 539
Expressed by non-central, recurrence relation of 516
Expressed by non-central, symmetric 514
Expressed by non-central, table of 517
Expressed by Stirling numbers of the second kind 519
Expressing power series 520
Expressing sample moments 521
Expressing sample moments, generating function of 518
Expressing sample moments, non-central 538
Faa di Bruno formula 420
Faa di Bruno formula, generalization 434
Factorial 43 104
Factorial convolution formula 105
Factorial generating function 216
Factorial moments of a sequence 215
Factorial, central 329
Factorial, falling 105
Factorial, rising 105
Factors of a positive integer, enumeration of 19 37
Family of elements 8
Fibonacci numbers 74 100 101 253 269 401
Fibonacci numbers of order s 269
Fibonacci numbers, convolution of 271
Fibonacci numbers, generalized 270
Fibonacci numbers, generating function of 254
Fibonacci numbers, table of 254
Formation of committees 352 353
Frechet inequality 166
Function 7
Galilei problem 92 161
Gambler’s ruin 242
Gambler’s ruin, expected time to 246
Gauss — Jacobi identity 394 402
Gegenbauer polynomials 450
Generalized factorial coefficients 152 165 307 312 344 345 347
Generalized factorial coefficients in terms of Stirling numbers 304
Generalized factorial coefficients, associated 166 326 327 366
Generalized factorial coefficients, asymptotic expression of 323
Generalized factorial coefficients, definition of 302
Generalized factorial coefficients, explicit expression of 306
Generalized factorial coefficients, generating function of 305
Generalized factorial coefficients, limiting expression of 307
Generalized factorial coefficients, non-central 318 319 334 540
Generalized factorial coefficients, orthogonality relation 308
Generalized factorial coefficients, recurrence relation of 309 310
Generalized factorial coefficients, table of 311
Generating function for the number of combinations 193 209
Generating function for the number of permutations 193 210
Generating function, binomial 201 216
Generating function, exponential 194
Generating function, factorial 201 216
Generating function, general form of 200
Generating function, moment 216
Generating function, multivariate 219
Generating function, ordinary 194
Genocchi numbers 544
Geometric progression, sum of terms of 21 33
Gould polynomials 207
Gould — Hopper numbers 318
Gould’s formula 208
Group of permutations 489
Group of permutations, alternating 490
Group of permutations, cycle indicator of 489
Group of permutations, cyclic 491
Group of permutations, identity 489
Group of permutations, symmetric 489 490
Gumbel inequality 166
Hanoi tower, transfer of 238
Hermite polynomials 449
Hermite polynomials, generalized 449
Homogeneous product sum symmetric functions 432 458
Inclusion and exclusion principle 134 136
Inclusion and exclusion principle, general expression of 145
Inequalities of Bonferroni 154
Inequality of Boole 155
Inequality of Frechet 166
Inequality of Gumbel 166
Integer solutions of a linear equation 68
Integer solutions, distinguishable urns 71
Integer solutions, enumeration of 69—71 98 138
Inverse map 8
Inverse relation 7
Inverse relations 284
Inversion of power series 437 438
Inversions of permutations 541
Lagrange inversion formula 435
Lagrange series 205 439
Laguerre polynomials 449 452
Lah numbers 302
Lah numbers, non-central 335
Lah numbers, signless 302
Lattice paths 75
Lattice paths with diagonal steps 100
Lattice paths, enumeration of 76—78
Lattice paths, figure 77
Leibnitz numbers 127 275
Lucas numbers 75 270
Lucas numbers of order s 270
Lucas numbers, generalized 271
Maclaurin power series 202
Magic squares 1
MAP 7
Map, bijective 7
Map, identity 8
Map, injective 7
Map, inverse 8
Map, surjective 7
Mean of a sequence 215
Menages numbers, reduced 142
Menages numbers, table of 142
Menages problem 140
Menages problem, modified 163
Menages problem, straight table 163
Mobius function 166
Mobius inversion formula 167
Models of necklaces 498 511
Moment generating function 216
Moments of a sequence 215
Money-changing problem 383
Monomials in a polynomial, enumeration of 99
Morse code 36 346
Multinomial coefficient(s) 123
Multinomial coefficient(s), generating function of 222
Multinomial coefficient(s), recurrence relation of 64
Multinomial coefficient(s), sum of 150 165 400
Multinomial formula 123
Multiple coincidences 187
Multiplication principle 20
Multivariate generating function 219
Newcomb’s problem 513
Newton factorial series 204
Newton’s formula, binomial 111
Newton’s formula, general binomial 115
Newton’s formula, negative binomial 112
Noerlund — Bernoulli numbers 328
Null set 3
Number system of base 5 159
Number system, binary 19 54
Number system, quaternary 224
Number system, ternary 47
Occupancy problem 339
Occupancy problem, classical 340
Operator 286
Operator 286
Orbit 493
Orbit, enumeration of the elements of 494
Order relation 7
Ordered n-tuple 5
Ordered pair 5
Parcelling out procedures 228
Partial derivatives of a function, number of 99
Partial sums, generating function of 255
Partition (s) of a finite set definition of 14
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