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 abstra
Author(s): Bruce Eli Sagan
Series: Graduate Studies in Mathematics 210
Publisher: AMS
Year: 2020
Language: English
Commentary: decrypted from 1590975CF33E19DA768D956D8DE9D996 source file
Pages: 304
Cover
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 involutions
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
Exercises
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 function
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 posets
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
Index
Back Cover
