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E-raamat: Enumerative Combinatorics

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Enumerative CombinatoricsSuurem pilt
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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.
Preface v
Basic Counting Principles
1(38)
Introduction
1(2)
Sets, relations and maps
3(11)
Basic notions
3(2)
Cartesian product
5(2)
Relations
7(1)
Maps
7(2)
Countable and uncountable sets
9(1)
Set operations
9(4)
Divisions and partitions of a set
13(1)
The principles of addition and multiplication
14(10)
Discrete probability
24(3)
Sums and products
27(8)
Bibliographic notes
35(1)
Exercises
36(3)
Permutations and Combinations
39(64)
Introduction
39(1)
Permutations
40(11)
Combinations
51(11)
Divisions and partitions of a finite set
62(6)
Integer solutions of a linear equation
68(7)
Lattice paths
75(7)
Probabilistic applications
82(8)
Classical problems in discrete probability
82(4)
Ordered and unordered samples
86(3)
Probability models in statistical mechanics
89(1)
Bibliographic notes
90(1)
Exercises
91(12)
Factorials, Binomial and Multinomial Co-Efficients
103(28)
Introduction
103(1)
Factorials
104(6)
Binomial coefficients
110(13)
Multinomial coefficients
123(1)
Bibliographic notes
124(1)
Exercises
124(7)
The Principle of Inclusion and Exclusion
131(38)
Introduction
131(1)
Number of elements in a union of sets
132(12)
Number of elements in a given number of sets
144(8)
Bonferroni inequalities
152(3)
Number of elements of a given rank
155(3)
Bibliographic notes
158(1)
Exercises
159(10)
Permutations with Fixed Points and Successions
169(22)
Introduction
169(1)
Permutations with fixed points
169(5)
Ranks of permutations
174(2)
Permutations with successions
176(4)
Circular permutations with successions
180(4)
Bibliographic notes
184(1)
Exercises
184(7)
Generating Functions
191(42)
Introduction
191(1)
Univariate generating functions
192(16)
Definitions and basic properties
192(10)
Power, factorial and Lagrange series
202(6)
Combinations and permutations
208(7)
Moment generating functions
215(4)
Multivariate generating functions
219(4)
Bibliographic notes
223(1)
Exercises
223(10)
Recurrence Relations
233(44)
Introduction
233(1)
Basic notions
233(2)
Recurrence relations of the first order
235(4)
The method of characteristic roots
239(11)
The method of generating functions
250(14)
Bibliographic notes
264(1)
Exercises
264(13)
Stirling Numbers
277(62)
Introduction
277(1)
Stirling numbers of the first and second kind
278(11)
Explicit expressions and recurrence relations
289(12)
Generalized factorial coefficients
301(13)
Non-central Stirling and related numbers
314(5)
Bibliographic notes
319(2)
Exercises
321(18)
Distributions and Occupancy
339(30)
Introduction
339(1)
Classical occupancy and modifications
340(8)
Ordered distributions and occupancy
348(2)
Balls of general specification and distinguishable urns
350(3)
Generating functions
353(5)
Bibliographic notes
358(1)
Exercises
359(10)
Partitions of Integers
369(42)
Introduction
369(1)
Recurrence relations and generating functions
370(6)
A universal generating function
376(7)
Interrelations among partition numbers
383(8)
Combinatorial identities
391(5)
Bibliographic notes
396(1)
Exercises
396(15)
Partition Polynomials
411(50)
Introduction
411(1)
Exponential Bell partition polynomials
412(7)
General partition polynomials
419(5)
Logarithmic partition polynomials
424(4)
Potential partition polynomials
428(5)
Inversion of power series
433(9)
Touchard polynomials
442(5)
Bibliographic notes
447(1)
Exercises
448(13)
Cycles of Permutations
461(26)
Introduction
461(1)
Permutations with a given number of cycles
462(6)
Even and odd permutations
468(3)
Permutations with partially ordered cycles
471(7)
Bibliographic notes
478(1)
Exercises
478(9)
Equivalence Classes
487(26)
Introduction
487(1)
Cycle indicator of a permutation group
488(5)
Orbits of elements of a finite set
493(6)
Models of colorings of a finite set
499(8)
Bibliographic notes
507(1)
Exercises
507(6)
Runs of Permutations and Eulerian Numbers
513(32)
Introduction
513(1)
Eulerian numbers
513(9)
Carlitz numbers
522(8)
Permutations with a given number of runs
530(3)
Permutations with repetition and a given number of runs
533(4)
Bibliographic notes
537(1)
Exercises
538(7)
Hints and Answers to Exercises 545(46)
Bibliography 591(10)
Index 601


Charalambides, Charalambos A.
Lisainfo e-raamatute kohta Lisainfo e-raamatute kohta
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