Introduction to Combinatorics

 
 
CRC Press
  • 2. Auflage
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  • erscheint ca. am 31. Dezember 2023
 
  • Buch
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  • Hardcover
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  • 424 Seiten
978-1-138-58262-0 (ISBN)
 
What Is Combinatorics Anyway?





Broadly speaking, combinatorics is the branch of mathematics dealing


with different ways of selecting objects from a set or arranging objects. It


tries to answer two major kinds of questions, namely, counting questions: how many ways can a selection or arrangement be chosen with a particular set of properties; and structural


questions: does there exist a selection or arrangement of objects with a


particular set of properties?





The authors have presented a text for students at all levels of preparation.


For some, this will be the first course where the students see several real proofs.


Others will have a good background in linear algebra, will have completed the calculus


stream, and will have started abstract algebra.





The text starts by briefly discussing several examples of typical combinatorial problems


to give the reader a better idea of what the subject covers. The next


chapters explore enumerative ideas and also probability. It then moves on to


enumerative functions and the relations between them, and generating functions and recurrences.,





Important families of functions, or numbers and then theorems are presented.


Brief introductions to computer algebra and group theory come next. Structures of particular


interest in combinatorics: posets, graphs, codes, Latin squares, and experimental designs follow. The


authors conclude with further discussion of the interaction between linear algebra


and combinatorics.





Features











Two new chapters on probability and posets.







Numerous new illustrations, exercises, and problems.







More examples on current technology use







A thorough focus on accuracy







Three appendices: sets, induction and proof techniques, vectors and matrices, and biographies with historical notes,







Flexible use of MapleTM and MathematicaTM
2nd New edition
  • Englisch
  • London
  • |
  • Großbritannien
Taylor & Francis Ltd
  • Für höhere Schule und Studium
  • Neue Ausgabe
105 Tables, black and white; 214 Illustrations, black and white
  • Höhe: 229 mm
  • |
  • Breite: 152 mm
978-1-138-58262-0 (9781138582620)

weitere Ausgaben werden ermittelt
W.D. Wallis is Professor Emeritus of Southern Illiniois University. John C George is Asscoiate Professor at Gordon State College.
Introduction


Some Combinatorial Examples


Sets, Relations and Proof Techniques


Two Principles of Enumeration


Graphs


Systems of Distinct Representatives


Fundamentals of Enumeration


Permutations and Combinations


Applications of P(n, k) and (n k)


Permutations and Combinations of Multisets


Applications and Subtle Errors


Algorithms


Probability


Introduction


Some Definitions and Easy Examples


Events and Probabilities


Three Interesting Examples


Probability Models


Bernoulli Trials


The Probabilities in Poker


The Wild Card Poker Paradox


The Pigeonhole Principle and Ramsey's Theorem


The Pigeonhole Principle


Applications of the Pigeonhole Principle


Ramsey's Theorem - the Graphical Case


Ramsey Multiplicity


Sum-Free Sets


Bounds on Ramsey Numbers


The General Form of Ramsey's Theorem


The Principle of Inclusion and Exclusion


Unions of Events


The Principle


Combinations with Limited Repetitions


Derangements


Generating Functions and Recurrence Relations


Generating Functions


Recurrence Relations


From Generating Function to Recurrence


Exponential Generating Functions


Catalan, Bell and Stirling Numbers


Introduction


Catalan Numbers


Stirling Numbers of the Second Kind


Bell Numbers


Stirling Numbers of the First Kind


Computer Algebra and Other Electronic Systems


Symmetries and the Polya-Redfield Method


Introduction


Basics of Groups


Permutations and Colorings


An Important Counting Theorem


Polya and Redfield's Theorem


Partially-Ordered Sets


Introduction


Examples and Definitions


Bounds and lattices


Isomorphism and Cartesian products


Extremal set theory: Sperner's and Dilworth's theorems


Introduction to Graph Theory


Degrees


Paths and Cycles in Graphs


Maps and Graph Coloring


Further Graph Theory


Euler Walks and Circuits


Application of Euler Circuits to Mazes


Hamilton Cycles


Trees


Spanning Trees


Coding Theory


Errors; Noise


The Venn Diagram Code


Binary Codes; Weight; Distance


Linear Codes


Hamming Codes


Codes and the Hat Problem


Variable-Length Codes and Data Compression


Latin Squares


Introduction


Orthogonality


Idempotent Latin Squares


Partial Latin Squares and Subsquares


Applications


Balanced Incomplete Block Designs


Design Parameters


Fisher's Inequality


Symmetric Balanced Incomplete Block Designs


New Designs from Old


Difference Methods


Linear Algebra Methods in Combinatorics


Recurrences Revisited


State Graphs and the Transfer Matrix Method


Kasteleyn's Permanent Method


Appendix 1: Sets; Proof Techniques7


Appendix 2: Matrices and Vectors


Appendix 3: Some Combinatorial People






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