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An Introduction to Enumeration / by Alan Camina, Barry Lewis
Contributor(s): Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Springer Undergraduate Mathematics Series | SpringerLink BücherPublisher: London : Springer-Verlag London Limited, 2011Description: Online-Ressource (XII, 232p. 62 illus, digital)ISBN:- 9780857296009
- 511.6
- QA164-167.2
- QA164.8
Contents:
Summary: What Is Enumeration? -- Generating Functions Count -- Working with Generating Functions -- Permutation Groups -- Matrices, Sequences and Sums -- Group Actions and Counting -- Exponential Generating Functions -- Graphs -- partitions and Paths.Summary: Written for students taking a second or third year undergraduate course in mathematics or computer science, this book is the ideal companion to a course in enumeration. Enumeration is a branch of combinatorics where the fundamental subject matter is numerous methods of pattern formation and counting. An Introduction to Enumeration provides a comprehensive and practical introduction to this subject giving a clear account of fundamental results and a thorough grounding in the use of powerful techniques and tools. Two major themes run in parallel through the book, generating functions and group theory. The former theme takes enumerative sequences and then uses analytic tools to discover how they are made up. Group theory provides a concise introduction to groups and illustrates how the theory can be used to count the number of symmetries a particular object has. These enrich and extend basic group ideas and techniques. The authors present their material through examples that are carefully chosen to establish key results in a natural setting. The aim is to progressively build fundamental theorems and techniques. This development is interspersed with exercises that consolidate ideas and build confidence. Some exercises are linked to particular sections while others range across a complete chapter. Throughout, there is an attempt to present key enumerative ideas in a graphic way, using diagrams to make them immediately accessible. The development assumes some basic group theory, a familiarity with analytic functions and their power series expansion along with some basic linear algebra.PPN: PPN: 1650945507Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
""Preface""; ""Contents""; ""1. What Is Enumeration?""; ""1.1 Bijections, Permutations and Sequences""; ""1.1.1 Exercises""; ""1.2 The Pigeonhole Principle""; ""1.2.1 Exercises""; ""1.3 The Principle of Inclusion and Exclusion""; ""1.3.1 Exercises""; ""1.4 The Principle of Exhaustion""; ""1.4.1 Exercises""; ""2. Generating Functions Count""; ""2.1 Countingfrom Polynomials to Power Series""; ""2.1.1 Exercises""; ""2.2 Recurrence Relations and Enumeration""; ""2.2.1 Exercises""; ""2.3 Sequence to Generating Function""; ""2.3.1 Sequence to Generating Function""
""2.3.2 Recurrence to Generating Function""""2.3.3 Exercises""; ""2.4 Miscellaneous Exercises""; ""3. Working with Generating Functions""; ""3.1 Expanding Generating Functions""; ""3.1.1 Exercises""; ""3.2 It's the Denominator that Counts""; ""3.2.1 Exercises""; ""3.3 Solving Linear Homogeneous Recurrences of Degree 2""; ""3.3.1 Exercises""; ""3.4 Miscellaneous Exercises""; ""4. Permutation Groups""; ""4.1 Introduction to Groups""; ""4.1.1 Exercises""; ""4.2 The Symmetric Group""; ""4.2.1 Exercises""; ""4.3 Group Actions""; ""4.3.1 Exercises""; ""4.4 Counting Subgroups""; ""4.4.1 Exercises""
""4.5 Miscellaneous Exercises""""5. Matrices, Sequences and Sums""; ""5.1 Pascal's Triangle and Enumeration""; ""5.1.1 Exercises""; ""5.2 The Push-Button Lock Sequence""; ""5.2.1 Exercises""; ""5.3 Pascal's Triangle as a Matrix""; ""5.3.1 Exercises""; ""5.4 Operations on Generating Functions""; ""5.4.1 Multiplication by zk""; ""5.4.2 The Product of Two Generating Functions""; ""5.4.3 Partial Sums of a Sequence""; ""5.4.4 The zDzddz Operation""; ""5.4.5 Exercises""; ""5.5 Miscellaneous Exercises""; ""6. Group Actions and Counting""; ""6.1 The First Steps""; ""6.1.1 Exercises""
""6.2 Colourings and the Cycle Index""""6.2.1 Exercises""; ""6.3 Polya's Theorem""; ""6.3.1 Exercises""; ""6.4 Miscellaneous Exercises""; ""7. Exponential Generating Functions""; ""7.1 Another Generating Function""; ""7.1.1 Exercises""; ""7.2 Recurrence to egf""; ""7.2.1 Exercises""; ""7.3 Operations on egfs""; ""7.3.1 Differentiation""; ""7.3.2 Products of Exponential Generating Functions""; ""7.3.3 Expanding egfs""; ""7.3.4 Exercises""; ""7.4 Counting Fixed Points in Permutations""; ""7.5 Counting more Permutations""; ""7.5.1 Counting Involutions""; ""7.5.2 Counting Zig-Zag Permutations""
""7.5.3 Exercises""""7.6 Miscellaneous Exercises""; ""8. Graphs""; ""8.1 Introduction to Graphs""; ""8.1.1 Exercises""; ""8.2 Connectivity""; ""8.2.1 Exercises""; ""8.3 Counting Graphs and Trees""; ""8.3.1 Counting Labelled Graphs and Trees""; ""8.3.2 Counting Unlabelled Graphs""; ""8.3.3 Exercises""; ""8.4 Planarity""; ""8.4.1 Exercises""; ""8.5 Miscellaneous Exercises""; ""9. Partitions and Paths""; ""9.1 Introducing Partitions""; ""9.1.1 Unrestricted Partitions""; ""9.1.2 Distinct Partitions""; ""9.1.3 Odd Partitions""; ""9.1.4 Relations Between Different Partitions""
""9.1.5 Ferrers Diagram""
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