A walk through combinatorics: an introduction to enumeration and graph theory
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| Format: | Book |
| Language: | English |
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New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo
World Scientific
[2017]
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| Edition: | Fourth edition |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Inhaltsverzeichnis |
| Item Description: | Enthält Literaturverzeichnis (Seite 583-586) und Register (Seite 587-593) |
| Physical Description: | xx, 593 Seiten Diagramme |
| ISBN: | 9789813148840 |
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| 245 | 1 | 0 | |a A walk through combinatorics |b an introduction to enumeration and graph theory |c Miklós Bóna (University of Florida, USA) ; with a foreword by Richard Stanley |
| 250 | |a Fourth edition | ||
| 264 | 1 | |a New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo |b World Scientific |c [2017] | |
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Record in the Search Index
| _version_ | 1804177552792616960 |
|---|---|
| adam_text | Titel: A walk through combinatorics
Autor: Bóna, Miklós
Jahr: 2017
Contents
Foreword vii
Preface ix
Acknowledgments xi
I. Basic Methods
1. Seven Is More Than Six. The Pigeon-Hole Principle 1
1.1 The Basic Pigeon-Hole Principle ............................1
Quick Check....................................................3
1.2 The Generalized Pigeon-Hole Principle......................4
Quick Chek....................................................10
Exercises..............................................................10
Supplementary Exercises............................................12
Solutions to Exercises................................................14
2. One Step at a Time. The Method of Mathematical Induction 23
2.1 Weak Induction................................................23
Quick Check....................................................28
2.2 Strong Induction..............................................29
Quick Check....................................................30
Exercises..............................................................31
Supplementary Exercises............................................33
Solutions to Exercises................................................35
II. Enumerative Combinatorics
3. There Are A Lot Of Them. Elementary Counting Problems 43
43
3.1 Permutations.........................
Quick Check....................................................46
3.2 Strings over a Finite Alphabet................................46
Quick Check....................................................50
3.3 Choice Problems..............................................50
Quick Check....................................................53
Exercises..............................................................54
Supplementary Exercises............................................58
Solutions to Exercises................................................60
4. No Matter How You Slice It. The Binomial Theorem and
Related Identities 73
4.1 The Binomial Theorem........................................73
Quick Check....................................................78
4.2 The Multinomial Theorem....................................78
Quick Check....................................................81
4.3 When the Exponent Is Not a Positive Integer..............81
Quick Check....................................................83
Exercises..............................................................83
Supplementary Exercises............................................87
Solutions to Exercises................................................90
5. Divide and Conquer. Partitions 101
5.1 Compositions..................................................101
Quick Check....................................................103
5.2 Set Partitions..................................................103
Quick Check....................................................106
5.3 Integer Partitions..............................................106
Quick Check....................................................112
Exercises..............................................213
Supplementary Exercises............................................215
Solutions to Exercises......................................217
6. Not So Vicious Cycles. Cycles in Permutations 123
6.1 Cycles in Permutations..................................124
Quick Check....................................................130
6.2 Permutations with Restricted Cycle Structure..............130
Quick Check....................................................134
Exercises..............................................................135
Supplementary Exercises............................................137
Solutions to Exercises................................................140
7. You Shall Not Overcount. The Sieve 147
7.1 Enumerating The Elements of Intersecting Sets............147
Quick Check....................................................150
7.2 Applications of the Sieve Formula............................150
Quick Check....................................................154
Exercises..............................................................155
Supplementary Exercises............................................156
Solutions to Exercises................................................157
8. A Function Is Worth Many Numbers. Generating Functions 163
8.1 Ordinary Generating Functions..............................163
8.1.1 Recurrence Relations and Generating Functions . 163
8.1.2 Products of Generating Functions..................169
8.1.3 Compositions of Generating Functions..............176
Quick Check....................................................180
8.2 Exponential Generating Functions ..........................180
8.2.1 Recurrence Relations and Exponential Generating
Functions..............................................180
8.2.2 Products of Exponential Generating Functions . . 182
8.2.3 Compositions of Exponential Generating Functions 185
Quick Check....................................................189
Exercises..............................................................189
Supplementary Exercises............................................191
Solutions to Exercises................................................195
III. Graph Theory
9. Dots and Lines. The Origins of Graph Theory 205
9.1 The Notion of Graphs. Eulerian Trails......................205
Quick Check....................................................210
9.2 Hamiltonian Cycles............................................210
Quick Check..........................212
9.3 Directed Graphs .......................213
Quick Check..........................215
9.4 The Notion of Isomorphisms 216
Quick Check......... 218
Exercises.............. 219
Supplementary Exercises..... 222
Solutions to Exercises....... 225
10. Staying Connected. Trees 233
10.1 Minimally Connected Graphs................................233
Quick Check....................................................239
10.2 Minimum-weight Spanning Trees. KruskaTs Greedy Algo-
rithm ............................................................239
Quick Check....................................................243
10.3 Graphs and Matrices..........................................244
10.3.1 Adjacency Matrices of Graphs......................244
Quick Check....................................................247
10.4 The Number of Spanning Trees of a Graph ................247
Quick Check....................................................253
Exercises..............................................................253
Supplementary Exercises............................................256
Solutions to Exercises................................................258
11. Finding A Good Match. Coloring and Matching 269
11.1 Introduction....................................................269
Quick Check....................................................271
11.2 Bipartite Graphs..............................................271
Quick Check....................................................276
11.3 Matchings in Bipartite Graphs ..............................277
11.3.1 Bipartite Graphs with Perfect Matchings..........279
11.3.2 Stable Matchings in Bipartite Graphs..............283
Quick Check....................................................285
11.4 More Than Two Colors ......................................285
Quick Check....................................................287
11.5 Matchings in Graphs That Are Not Bipartite..............287
Quick Check....................................................291
Exercises................................................291
Supplementary Exercises...................... 293
Solutions to Exercises........................ 295
12. Do Not Cross. Planar Graphs 301
12.1 Euler s Theorem for Planar Graphs..........................301
Quick Check.......................... 304
12.2 Polyhedra......................................................304
Quick Check....................................................311
12.3 Coloring Maps ................................................311
Quick Check....................................................313
Exercises..........................................314
Supplementary Exercises...................................315
Solutions to Exercises............................317
IV. Horizons
13. Does It Clique? Ramsey Theory 321
13.1 Ramsey Theory for Finite Graphs............................321
Quick Check....................................................327
13.2 Generalizations of the Ramsey Theorem....................327
Quick Check....................................................330
13.3 Ramsey Theory in Geometry................................331
Quick Check....................................................333
Exercises..............................................................334
Supplementary Exercises............................................335
Solutions to Exercises................................................337
14. So Hard To Avoid. Subsequence Conditions on Permutations 343
14.1 Pattern Avoidance............................................343
Quick Check....................................................352
14.2 Stack Sortable Permutations..................................353
Quick Check....................................................363
Exercises.............................................364
Supplementary Exercises............................................366
Solutions to Exercises................................................368
15. Who Knows What It Looks Like, But It Exists. The
Probabilistic Method 381
15.1 The Notion of Probability....................................381
Quick Check....................................................384
15.2 Non-constructive Proofs......................................385
Quick Check....................................................387
15.3 Independent Events............................................387
15.3.1 The Notion of Independence and Bayes Theorem 387
15.3.2 More Than Two Events..............................392
Quick Check....................................................394
15.4 Expected Values ..............................................395
15.4.1 Linearity of Expectation ............................396
15.4.2 Existence Proofs Using Expectation................399
15.4.3 Conditional Expectation ............................401
Quick Check....................................................403
Exercises..............................................................403
Supplementary Exercises............................................406
Solutions to Exercises................................................410
16. At Least Some Order. Partial Orders and Lattices 417
16.1 The Notion of Partially Ordered Set........................417
Quick Check....................................................423
16.2 The Mobius Function of a Poset..............................423
Quick Check....................................................431
16.3 Lattices........................................................431
Quick Check....................................................438
Exercises..............................................................438
Supplementary Exercises............................................440
Solutions to Exercises................................................443
17. As Evenly As Possible. Block Designs and Error Cor-
recting Codes 451
17.1 Introduction....................................................451
17.1.1 Moto-cross Races ....................................451
17.1.2 Incompatible Computer Programs..................453
Quick Check....................................................455
17.2 Balanced Incomplete Block Designs........................455
Quick Check..............................................458
17.3 New Designs From Old..................................463
Quick Check..............
17.4 Existence of Certain BIBDs................. 463
17.4.1 A Residual Design of a Projective Plane...... 465
Quick Check.......................... 466
17.5 Codes and Designs...................... 466
17.5.1 Coding Theory........................................466
17.5.2 Error Correcting Codes..............................467
17.5.3 Formal Definitions About Codes....................468
17.5.4 Perfect Codes ........................................472
Quick Check....................................................475
Exercises..............................................................476
Supplementary Exercises............................................478
Solutions to Exercises................................................479
18. Are They Really Different? Counting Unlabeled Structures 487
18.1 Enumeration Under Group Action ..........................487
18.1.1 Introduction..........................................487
18.1.2 Groups................................................487
18.1.3 Permutation Groups..................................490
Quick Check....................................................497
18.2 Counting Unlabeled Trees....................................498
18.2.1 Counting Rooted Non-plane 1-2 Trees..............498
18.2.2 Counting Rooted Non-plane Trees..................501
18.2.3 Counting Unrooted Trees............................503
Quick Check....................................................509
Exercises..............................................................510
Supplementary Exercises............................................513
Solutions to Exercises................................................515
19. The Sooner The Better. Combinatorial Algorithms 523
19.1 In Lieu of Definitions..........................................523
19.1.1 The Halting Problem................................524
Quick Check....................................................525
19.2 Sorting Algorithms............................................526
19.2.1 BubbleSort............................................526
19.2.2 MergeSort ............................................529
19.2.3 Comparing the Growth of Functions................532
Quick Check....................................................534
19.3 Algorithms on Graphs........................................534
19.3.1 Minimum-cost Spanning Trees, Revisited..........534
19.3.2 Finding the Shortest Path ..........................537
Quick Check....................................................542
Exercises..............................................................543
Supplementary Exercises............................................546
Solutions to Exercises................................................547
20. Does Many Mean More Than One? Computational Complexity 553
20.1 Turing Machines..............................................553
Quick Check....................................................556
20.2 Complexity Classes............................................556
20.2.1 The Class P..........................................556
20.2.2 The Class NP........................................558
20.2.3 NP-complete Problems..............................565
20.2.4 Other Complexity Classes............................571
Quick Check....................................................574
Exercises..............................................................574
Supplementary Exercises............................................576
Solutions to Exercises................................................577
Bibliography 583
Index 587
|
| any_adam_object | 1 |
| author | Bóna, Miklós |
| author_GND | (DE-588)1013929861 |
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| discipline | Mathematik |
| edition | Fourth edition |
| format | Book |
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| indexdate | 2024-07-10T07:49:54Z |
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| spelling | Bóna, Miklós Verfasser (DE-588)1013929861 aut A walk through combinatorics an introduction to enumeration and graph theory Miklós Bóna (University of Florida, USA) ; with a foreword by Richard Stanley Fourth edition New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo World Scientific [2017] © 2017 xx, 593 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Enthält Literaturverzeichnis (Seite 583-586) und Register (Seite 587-593) Kombinatorische Analysis (DE-588)4164746-4 gnd rswk-swf Ramsey-Theorie (DE-588)4212682-4 gnd rswk-swf Kombinatorik (DE-588)4031824-2 gnd rswk-swf Abzählende Kombinatorik (DE-588)4132720-2 gnd rswk-swf Graphentheorie (DE-588)4113782-6 gnd rswk-swf Combinatorial analysis Textbooks Combinatorial enumeration problems Textbooks Graph theory Textbooks Abzählende Kombinatorik (DE-588)4132720-2 s DE-604 Kombinatorische Analysis (DE-588)4164746-4 s Kombinatorik (DE-588)4031824-2 s Ramsey-Theorie (DE-588)4212682-4 s Graphentheorie (DE-588)4113782-6 s 1\p DE-604 DE-601 application/pdf http://www.gbv.de/dms/tib-ub-hannover/862823498.pdf Inhaltsverzeichnis HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029732236&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Bóna, Miklós A walk through combinatorics an introduction to enumeration and graph theory Kombinatorische Analysis (DE-588)4164746-4 gnd Ramsey-Theorie (DE-588)4212682-4 gnd Kombinatorik (DE-588)4031824-2 gnd Abzählende Kombinatorik (DE-588)4132720-2 gnd Graphentheorie (DE-588)4113782-6 gnd |
| subject_GND | (DE-588)4164746-4 (DE-588)4212682-4 (DE-588)4031824-2 (DE-588)4132720-2 (DE-588)4113782-6 |
| title | A walk through combinatorics an introduction to enumeration and graph theory |
| title_auth | A walk through combinatorics an introduction to enumeration and graph theory |
| title_exact_search | A walk through combinatorics an introduction to enumeration and graph theory |
| title_full | A walk through combinatorics an introduction to enumeration and graph theory Miklós Bóna (University of Florida, USA) ; with a foreword by Richard Stanley |
| title_fullStr | A walk through combinatorics an introduction to enumeration and graph theory Miklós Bóna (University of Florida, USA) ; with a foreword by Richard Stanley |
| title_full_unstemmed | A walk through combinatorics an introduction to enumeration and graph theory Miklós Bóna (University of Florida, USA) ; with a foreword by Richard Stanley |
| title_short | A walk through combinatorics |
| title_sort | a walk through combinatorics an introduction to enumeration and graph theory |
| title_sub | an introduction to enumeration and graph theory |
| topic | Kombinatorische Analysis (DE-588)4164746-4 gnd Ramsey-Theorie (DE-588)4212682-4 gnd Kombinatorik (DE-588)4031824-2 gnd Abzählende Kombinatorik (DE-588)4132720-2 gnd Graphentheorie (DE-588)4113782-6 gnd |
| topic_facet | Kombinatorische Analysis Ramsey-Theorie Kombinatorik Abzählende Kombinatorik Graphentheorie |
| url | http://www.gbv.de/dms/tib-ub-hannover/862823498.pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029732236&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT bonamiklos awalkthroughcombinatoricsanintroductiontoenumerationandgraphtheory |
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