Combinatorics : the art of counting / Bruce E. Sagan

By:

Sagan, Bruce E, 1954- [VerfasserIn]

Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Graduate Studies in Mathematics Ser ; v.210Publisher: Providence, Rhode Island : American Mathematical Society, [2020]Description: 1 Online-Ressource (1 online resource)ISBN:

147046280X
9781470462802

Subject(s):

Genre/Form:

Lehrbuch

Additional physical formats: 9781470460327 | 1470460327 | Erscheint auch als: Combinatorics. Druck-Ausgabe Providence, Rhode Island : American Mathematical Society, 2020. xix, 304 SeitenDDC classification:

511.6

MSC: MSC: 05-01 | 06-01RVK: RVK: SK 170LOC classification:

QA164

Online resources:

Zugang im Netz des KIT

Summary: Intro -- Title page -- Copyright -- Contents -- Preface -- List of Notation -- Chapter 1. Basic Counting -- 1.1. The Sum and Product Rules for sets -- 1.2. Permutations and words -- 1.3. Combinations and subsets -- 1.4. Set partitions -- 1.5. Permutations by cycle structure -- 1.6. Integer partitions -- 1.7. Compositions -- 1.8. The twelvefold way -- 1.9. Graphs and digraphs -- 1.10. Trees -- 1.11. Lattice paths -- 1.12. Pattern avoidance -- Exercises -- Chapter 2. Counting with Signs -- 2.1. The Principle of Inclusion and Exclusion -- 2.2. Sign-reversing involutionsSummary: 2.3. The Garsia-Milne Involution Principle -- 2.4. The Reflection Principle -- 2.5. The Lindström-Gessel-Viennot Lemma -- 2.6. The Matrix-Tree Theorem -- Exercises -- Chapter 3. Counting with Ordinary Generating Functions -- 3.1. Generating polynomials -- 3.2. Statistics and -analogues -- 3.3. The algebra of formal power series -- 3.4. The Sum and Product Rules for ogfs -- 3.5. Revisiting integer partitions -- 3.6. Recurrence relations and generating functions -- 3.7. Rational generating functions and linear recursions -- 3.8. Chromatic polynomials -- 3.9. Combinatorial reciprocity -- ExercisesSummary: Chapter 4. Counting with Exponential Generating Functions -- 4.1. First examples -- 4.2. Generating functions for Eulerian polynomials -- 4.3. Labeled structures -- 4.4. The Sum and Product Rules for egfs -- 4.5. The Exponential Formula -- Exercises -- Chapter 5. Counting with Partially Ordered Sets -- 5.1. Basic properties of partially ordered sets -- 5.2. Chains, antichains, and operations on posets -- 5.3. Lattices -- 5.4. The Möbius function of a poset -- 5.5. The Möbius Inversion Theorem -- 5.6. Characteristic polynomials -- 5.7. Quotients of posets -- 5.8. Computing the Möbius functionSummary: 5.9. Binomial posets -- Exercises -- Chapter 6. Counting with Group Actions -- 6.1. Groups acting on sets -- 6.2. Burnside's Lemma -- 6.3. The cycle index -- 6.4. Redfield-Pólya theory -- 6.5. An application to proving congruences -- 6.6. The cyclic sieving phenomenon -- Exercises -- Chapter 7. Counting with Symmetric Functions -- 7.1. The algebra of symmetric functions, Sym -- 7.2. The Schur basis of Sym -- 7.3. Hooklengths -- 7.4. -partitions -- 7.5. The Robinson-Schensted-Knuth correspondence -- 7.6. Longest increasing and decreasing subsequences -- 7.7. Differential posetsSummary: 7.8. The chromatic symmetric function -- 7.9. Cyclic sieving redux -- Exercises -- Chapter 8. Counting with Quasisymmetric Functions -- 8.1. The algebra of quasisymmetric functions, QSym -- 8.2. Reverse -partitions -- 8.3. Chain enumeration in posets -- 8.4. Pattern avoidance and quasisymmetric functions -- 8.5. The chromatic quasisymmetric function -- Exercises -- Appendix. Introduction to Representation Theory -- A.1. Basic notions -- Exercises -- Bibliography -- IndexSummary: This book is a gentle introduction to the enumerative part of combinatorics suitable for study at the advanced undergraduate or beginning graduate level. In addition to covering all the standard techniques for counting combinatorial objects, the text contains material from the research literature which has never before appeared in print, such as the use of quotient posets to study the Möbius function and characteristic polynomial of a partially ordered set, or the connection between quasisymmetric functions and pattern avoidance.The book assumes minimal background, and a first course in abstraPPN: PPN: 1755153538Package identifier: Produktsigel: BSZ-4-NLEBK-KAUB | ZDB-4-NLEBK

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