Introduction to Enumerative and Analytic Combinatorics

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Opis: Introduction to Enumerative and Analytic Combinatorics - Miklos Bona

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. "Miklos Bona has done a masterful job of bringing an overview of all of enumerative combinatorics within reach of undergraduates. The two fundamental themes of bijective proofs and generating functions, together with their intimate connections, recur constantly. A wide selection of topics, including several never appearing before in a textbook, is included that gives an idea of the vast range of enumerative combinatorics. In particular, for those with sufficient background in undergraduate linear algebra and abstract algebra, there are many tantalizing hints of the fruitful connection between enumerative combinatorics and algebra that play a central role in the subject of algebraic combinatorics." -From the Foreword to the First Edition by Richard Stanley, Cambridge, Massachusetts, USAMETHODS Basic methods When we add and when we subtract When we multiply When we divide Applications of basic counting principles The pigeonhole principle Notes Chapter review Exercises Solutions to exercises Supplementary exercises Applications of basic methods Multisets and compositions Set partitions Partitions of integers The inclusion-exclusion principle The twelvefold way Notes Chapter review Exercises Solutions to exercises Supplementary exercises Generating functions Power series Warming up: Solving recurrence relations Products of generating functions Compositions of generating functions A different type of generating functions Notes Chapter review Exercises Solutions to exercises Supplementary exercises TOPICS Counting permutations Eulerian numbers The cycle structure of permutations Cycle structure and exponential generating functions Inversions Advanced applications of generating functions to permutation enumeration Notes Chapter review Exercises Solutions to exercises Supplementary exercises Counting graphs Trees and forests Graphs and functions When the vertices are not freely labeled Graphs on colored vertices Graphs and generating functions Notes Chapter review Exercises Solutions to exercises Supplementary exercises Extremal combinatorics Extremal graph theory Hypergraphs Something is more than nothing: Existence proofs Notes Chapter review Exercises Solutions to exercises Supplementary exercises AN ADVANCED METHOD Analytic combinatorics Exponential growth rates Polynomial precision More precise asymptotics Notes Chapter review Exercises Solutions to exercises Supplementary exercises SPECIAL TOPICS Symmetric structures Designs Finite projective planes Error-correcting codes Counting symmetric structures Notes Chapter review Exercises Solutions to exercises Supplementary exercises Sequences in combinatorics Unimodality Log-concavity The real zeros property Notes Chapter review Exercises Solutions to exercises Supplementary exercises Counting magic squares and magic cubes A distribution problem Magic squares of fixed size Magic squares of fixed line sum Why magic cubes are different Notes Chapter review Exercises Solutions to exercises Supplementary exercises Appendix: The method of mathematical induction Weak induction Strong induction


Szczegóły: Introduction to Enumerative and Analytic Combinatorics - Miklos Bona

Tytuł: Introduction to Enumerative and Analytic Combinatorics
Autor: Miklos Bona
Producent: Apple
ISBN: 9781482249095
Rok produkcji: 2015
Ilość stron: 556
Oprawa: Twarda
Waga: 0.93 kg


Recenzje: Introduction to Enumerative and Analytic Combinatorics - Miklos Bona

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Introduction to Enumerative and Analytic Combinatorics

  • Producent: Apple
  • Oprawa: Twarda

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. "Miklos Bona has done a masterful job of bringing an overview of all of enumerative combinatorics within reach of undergraduates. The two fundamental themes of bijective proofs and generating functions, together with their intimate connections, recur constantly. A wide selection of topics, including several never appearing before in a textbook, is included that gives an idea of the vast range of enumerative combinatorics. In particular, for those with sufficient background in undergraduate linear algebra and abstract algebra, there are many tantalizing hints of the fruitful connection between enumerative combinatorics and algebra that play a central role in the subject of algebraic combinatorics." -From the Foreword to the First Edition by Richard Stanley, Cambridge, Massachusetts, USAMETHODS Basic methods When we add and when we subtract When we multiply When we divide Applications of basic counting principles The pigeonhole principle Notes Chapter review Exercises Solutions to exercises Supplementary exercises Applications of basic methods Multisets and compositions Set partitions Partitions of integers The inclusion-exclusion principle The twelvefold way Notes Chapter review Exercises Solutions to exercises Supplementary exercises Generating functions Power series Warming up: Solving recurrence relations Products of generating functions Compositions of generating functions A different type of generating functions Notes Chapter review Exercises Solutions to exercises Supplementary exercises TOPICS Counting permutations Eulerian numbers The cycle structure of permutations Cycle structure and exponential generating functions Inversions Advanced applications of generating functions to permutation enumeration Notes Chapter review Exercises Solutions to exercises Supplementary exercises Counting graphs Trees and forests Graphs and functions When the vertices are not freely labeled Graphs on colored vertices Graphs and generating functions Notes Chapter review Exercises Solutions to exercises Supplementary exercises Extremal combinatorics Extremal graph theory Hypergraphs Something is more than nothing: Existence proofs Notes Chapter review Exercises Solutions to exercises Supplementary exercises AN ADVANCED METHOD Analytic combinatorics Exponential growth rates Polynomial precision More precise asymptotics Notes Chapter review Exercises Solutions to exercises Supplementary exercises SPECIAL TOPICS Symmetric structures Designs Finite projective planes Error-correcting codes Counting symmetric structures Notes Chapter review Exercises Solutions to exercises Supplementary exercises Sequences in combinatorics Unimodality Log-concavity The real zeros property Notes Chapter review Exercises Solutions to exercises Supplementary exercises Counting magic squares and magic cubes A distribution problem Magic squares of fixed size Magic squares of fixed line sum Why magic cubes are different Notes Chapter review Exercises Solutions to exercises Supplementary exercises Appendix: The method of mathematical induction Weak induction Strong induction

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