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Charalambides C.A. — Enumerative Combinatorics
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Название: Enumerative CombinatoricsАвтор: 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.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 2002Количество страниц: 609Добавлена в каталог: 18.11.2009Операции: Положить на полку |
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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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