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E-raamat: Introduction to Enumerative and Analytic Combinatorics

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Introduction to Enumerative and Analytic CombinatoricsSuurem pilt
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Introduction to Enumerative and Analytic Combinatorics fills the gap between introductory texts in discrete mathematics and advanced graduate texts in enumerative combinatorics. The book first deals with basic counting principles, compositions and partitions, and generating functions. It then focuses on the structure of permutations, graph enumeration, and extremal combinatorics. Lastly, the text discusses supplemental topics, including error-correcting codes, properties of sequences, and magic squares.

Strengthening the analytic flavor of the book, this Second Edition:





Features a new chapter on analytic combinatorics and new sections on advanced applications of generating functions Demonstrates powerful techniques that do not require the residue theorem or complex integration Adds new exercises to all chapters, significantly extending coverage of the given topics

Introduction to Enumerative and Analytic Combinatorics, Second Edition makes combinatorics more accessible, increasing interest in this rapidly expanding field.

Outstanding Academic Title of the Year, Choice magazine, American Library Association.

Arvustused

Bona's work is a superb text for any reader learning the vast topic of combinatorics. It includes a well-written description of the fundamentals of combinatorics and several chapters of applications. Each chapter concludes with a list of important formulas available for future reference and a lengthy list of exercises. These exercises are quite comprehensive in that they include a wide range of topics, many exploring other interesting topics unexplained in the text. Most of these exercises are accompanied by complete, well-explained solutions to assist struggling readers. One of the best aspects of the book is the conversational tone in which it is written. When reading through the numerous proofs in the text, readers will feel as though they are actually in the classroom with Bona (Univ. of Florida). His explanations are clear and concise, and his dry humor is both entertaining and essential to the text's development. People spreading rumors, wearing colorful hats, and embarking on hazardous vacations are much more enjoyable to count than indistinguishable balls in jars. This work is an excellent addition to the combinatorics library.





--A. Misseldine, Southern Utah University

Foreword to the first edition xv
Preface to the second edition xvii
Acknowledgments xix
Frequently used notation xxi
I Methods 1(178)
1 Basic methods
3(52)
1.1 When we add and when we subtract
4(2)
1.1.1 When we add
4(1)
1.1.2 When we subtract
5(1)
1.2 When we multiply
6(8)
1.2.1 The product principle
6(3)
1.2.2 Using several counting principles
9(1)
1.2.3 When repetitions are not allowed
10(4)
1.2.3.1 Permutations
10(2)
1.2.3.2 Partial lists without repetition
12(2)
1.3 When we divide
14(5)
1.3.1 The division principle
14(2)
1.3.2 Subsets
16(3)
1.3.2.1 The number of k-element subsets of an n-element set
16(1)
1.3.2.2 The binomial theorem for positive integer exponents
17(2)
1.4 Applications of basic counting principles
19(13)
1.4.1 Bijective proofs
19(6)
1.4.1.1 Catalan numbers
23(2)
1.4.2 Properties of binomial coefficients
25(5)
1.4.3 Permutations with repetition
30(2)
1.5 The pigeonhole principle
32(4)
1.6 Notes
36(1)
1.7
Chapter review
37(1)
1.8 Exercises
38(4)
1.9 Solutions to exercises
42(7)
1.10 Supplementary exercises
49(6)
2 Applications of basic methods
55(62)
2.1 Multisets and compositions
56(3)
2.1.1 Weak compositions
56(2)
2.1.2 Compositions
58(1)
2.2 Set partitions
59(6)
2.2.1 Stirling numbers of the second kind
59(2)
2.2.2 Recurrence relations for Stirling numbers of the second kind
61(3)
2.2.3 When the number of blocks is not fixed
64(1)
2.3 Partitions of integers
65(12)
2.3.1 Nonincreasing finite sequences of positive integers
65(3)
2.3.2 Ferrers shapes and their applications
68(2)
2.3.3 Excursion: Euler's pentagonal number theorem
70(7)
2.4 The inclusion—exclusion principle
77(15)
2.4.1 Two intersecting sets
77(3)
2.4.2 Three intersecting sets
80(4)
2.4.3 Any number of intersecting sets
84(34)
2.4.3.1 An explicit formula for the numbers S(n, k)
85(2)
2.4.3.2 An application involving linear orders
87(1)
2.4.3.3 Euler's φ function
88(2)
2.4.3.4 Derangements
90(1)
2.4.3.5 Excursion: Another proof of the inclusion-exclusion principle
91(1)
2.5 The twelvefold way
92(3)
2.6 Notes
95(1)
2.7
Chapter review
96(1)
2.8 Exercises
97(4)
2.9 Solutions to exercises
101(10)
2.10 Supplementary exercises
111(6)
3 Generating functions
117(62)
3.1 Power series
118(4)
3.1.1 Generalized binomial coefficients
118(1)
3.1.2 Formal power series
119(3)
3.2 Warming up: Solving recurrence relations
122(9)
3.2.1 Ordinary generating functions
122(6)
3.2.2 Exponential generating functions
128(3)
3.3 Products of generating functions
131(16)
3.3.1 Ordinary generating functions
131(11)
3.3.2 Exponential generating functions
142(5)
3.4 Compositions of generating functions
147(13)
3.4.1 Ordinary generating functions
147(6)
3.4.2 Exponential generating functions
153(36)
3.4.2.1 The exponential formula
153(4)
3.4.2.2 The compositional formula
157(3)
3.5 A different type of generating functions
160(1)
3.6 Notes
161(1)
3.7
Chapter review
162(2)
3.8 Exercises
164(2)
3.9 Solutions to exercises
166(10)
3.10 Supplementary exercises
176(3)
II Topics 179(192)
4 Counting permutations
181(56)
4.1 Eulerian numbers
182(7)
4.2 The cycle structure of permutations
189(11)
4.2.1 Stirling numbers of the first kind
189(7)
4.2.2 Permutations of a given type
196(4)
4.3 Cycle structure and exponential generating functions
200(5)
4.4 Inversions
205(10)
4.4.1 Counting permutations with respect to inversions
210(5)
4.5 Advanced applications of generating functions to permutation enumeration
215(1)
4.5.1 The combinatorial meaning of the derivative
215(1)
4.5.2 Multivariate generating functions
215(1)
4.6 Notes
216(1)
4.7
Chapter review
217(1)
4.8 Exercises
218(4)
4.9 Solutions to exercises
222(10)
4.10 Supplementary exercises
232(5)
5 Counting graphs
237(78)
5.1 Trees and forests
241(18)
5.1.1 Trees
241(2)
5.1.2 The notion of graph isomorphisms
243(4)
5.1.3 Counting trees on labeled vertices
247(8)
5.1.4 Forests
255(4)
5.2 Graphs and functions
259(4)
5.2.1 Acyclic functions
259(1)
5.2.2 Parking functions
260(3)
5.3 When the vertices are not freely labeled
263(9)
5.3.1 Rooted plane trees
263(4)
5.3.2 Decreasing binary trees
267(5)
5.4 Graphs on colored vertices
272(11)
5.4.1 Chromatic polynomials
273(7)
5.4.2 Colored graphs
280(3)
5.4.2.1 Counting all k-colored graphs
280(1)
5.4.2.2 Counting colored trees
280(3)
5.5 Graphs and generating functions
283(10)
5.5.1 Trees counted by Cayley's formula
283(1)
5.5.2 Rooted trees
284(4)
5.5.2.1 Ordinary generating functions
284(1)
5.5.2.2 Exponential generating functions
285(1)
5.5.2.3 An application of multivariate generating functions
286(2)
5.5.3 Connected graphs
288(1)
5.5.4 Eulerian graphs
289(4)
5.6 Notes
293(2)
5.7
Chapter review
295(1)
5.8 Exercises
296(4)
5.9 Solutions to exercises
300(10)
5.10 Supplementary exercises
310(5)
6 Extremal combinatorics
315(56)
6.1 Extremal graph theory
315(19)
6.1.1 Bipartite graphs
316(4)
6.1.2 Turan's theorem
320(4)
6.1.3 Graphs excluding cycles
324(8)
6.1.3.1 Convex functions and Jensen's inequality
326(1)
6.1.3.2 Notation in approximate counting
327(1)
6.1.3.3 Refining the results on fC4(n)
328(2)
6.1.3.4 Avoiding longer cycles
330(2)
6.1.4 Graphs excluding complete bipartite graphs
332(2)
6.2 Hypergraphs
334(9)
6.2.1 Hypergraphs with pairwise intersecting edges
335(6)
6.2.1.1 Sunflowers
340(1)
6.2.2 Hypergraphs with pairwise incomparable edges
341(2)
6.3 Something is more than nothing: Existence proofs
343(7)
6.3.1 Property B
344(1)
6.3.2 Excluding monochromatic arithmetic progressions
345(1)
6.3.3 Codes over finite alphabets
346(4)
6.4 Notes
350(1)
6.5
Chapter review
350(1)
6.6 Exercises
351(5)
6.7 Solutions to exercises
356(12)
6.8 Supplementary exercises
368(3)
III An Advanced Method 371(46)
7 Analytic combinatorics
373(44)
7.1 Exponential growth rates
374(17)
7.1.1 Rational functions
374(6)
7.1.1.1 Motivation
374(1)
7.1.1.2 Theoretical background
374(6)
7.1.2 Singularity analysis
380(11)
7.1.2.1 Motivation
380(1)
7.1.2.2 Analytic functions
381(5)
7.1.2.3 The complex versions of some well-known functions
386(2)
7.1.2.4 Singularities
388(1)
7.1.2.5 The main theorem of exponential asymptotics
389(2)
7.2 Polynomial precision
391(5)
7.2.1 Rational functions again
391(5)
7.2.1.1 Motivation
392(1)
7.2.1.2 Multiple singularities in rational functions
393(3)
7.3 More precise asymptotics
396(4)
7.3.1 Entire functions divided by (1-x)
396(3)
7.3.2 Rational functions one more time
399(1)
7.4 Notes
400(1)
7.5
Chapter review
401(1)
7.6 Exercises
402(4)
7.7 Solutions to exercises
406(8)
7.8 Supplementary exercises
414(3)
IV Special Topics 417(104)
8 Symmetric structures
419(36)
8.1 Designs
419(7)
8.2 Finite projective planes
426(4)
8.2.1 Finite projective planes of prime power order
429(1)
8.3 Error-correcting codes
430(6)
8.3.1 Words far apart
430(3)
8.3.2 Codes from designs
433(1)
8.3.3 Perfect codes
433(3)
8.4 Counting symmetric structures
436(8)
8.5 Notes
444(1)
8.6
Chapter review
445(1)
8.7 Exercises
446(1)
8.8 Solutions to exercises
447(5)
8.9 Supplementary exercises
452(3)
9 Sequences in combinatorics
455(28)
9.1 Unimodality
455(3)
9.2 Log-concavity
458(10)
9.2.1 Log-concavity implies unimodality
458(3)
9.2.2 The product property
461(2)
9.2.3 Injective proofs
463(21)
9.2.3.1 Lattice paths again
466(2)
9.3 The real zeros property
468(4)
9.4 Notes
472(1)
9.5
Chapter review
472(1)
9.6 Exercises
473(2)
9.7 Solutions to exercises
475(5)
9.8 Supplementary exercises
480(3)
10 Counting magic squares and magic cubes
483(38)
10.1 A distribution problem
483(1)
10.2 Magic squares of fixed size
484(13)
10.2.1 The case of n = 3
485(3)
10.2.2 The function Hn (r) for fixed n
488(9)
10.3 Magic squares of fixed line sum
497(4)
10.4 Why magic cubes are different
501(4)
10.5 Notes
505(1)
10.6
Chapter review
506(1)
10.7 Exercises
507(2)
10.8 Solutions to exercises
509(10)
10.9 Supplementary exercises
519(2)
Appendix The method of mathematical induction 521(4)
A.1 Weak induction
521(1)
A.2 Strong induction
522(3)
Bibliography 525(6)
Index 531
Miklós Bóna received his Ph.D in mathematics from the Massachusetts Institute of Technology in 1997. Since 1999, he has taught at the University of Florida, where, in 2010, he was inducted into the Academy of Distinguished Teaching Scholars. Professor Bóna has mentored numerous graduate and undergraduate students. He is the author of four books and more than 65 research articles, mostly focusing on enumerative and analytic combinatorics. His book, Combinatorics of Permutations, won a 2006 Outstanding Title Award from Choice, the journal of the American Library Association. He is also an editor-in-chief for the Electronic Journal of Combinatorics, and for two book series at CRC Press.
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