**Graduate Studies in Mathematics**

Volume: 210;
2020;
304 pp;
Softcover

MSC: Primary 05;
Secondary 06

**Print ISBN: 978-1-4704-6032-7
Product Code: GSM/210**

List Price: $89.00

AMS Member Price: $71.20

MAA Member Price: $80.10

**Electronic ISBN: 978-1-4704-6280-2
Product Code: GSM/210.E**

List Price: $89.00

AMS Member Price: $71.20

MAA Member Price: $80.10

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#### Supplemental Materials

# Combinatorics: The Art of Counting

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*Bruce E. Sagan*

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 abstract
algebra should suffice. The exposition is very reader friendly:
keeping a moderate pace, using lots of examples, emphasizing recurring
themes, and frankly expressing the delight the author takes in
mathematics in general and combinatorics in particular.

#### Readership

Undergraduate and graduate students interested in combinatorics.

#### Table of Contents

# Table of Contents

## Combinatorics: The Art of Counting

- Title page 44
- Copyright 55
- Contents 88
- Preface 1212
- List of Notation 1414
- Chapter 1. Basic Counting 2222
- 1.1. The Sum and Product Rules for sets 2222
- 1.2. Permutations and words 2525
- 1.3. Combinations and subsets 2626
- 1.4. Set partitions 3131
- 1.5. Permutations by cycle structure 3232
- 1.6. Integer partitions 3434
- 1.7. Compositions 3737
- 1.8. The twelvefold way 3838
- 1.9. Graphs and digraphs 3939
- 1.10. Trees 4343
- 1.11. Lattice paths 4646
- 1.12. Pattern avoidance 4949
- Exercises 5454

- Chapter 2. Counting with Signs 6262
- Chapter 3. Counting with Ordinary Generating Functions 9292
- 3.1. Generating polynomials 9292
- 3.2. Statistics and 𝑞-analogues 9595
- 3.3. The algebra of formal power series 102102
- 3.4. The Sum and Product Rules for ogfs 107107
- 3.5. Revisiting integer partitions 110110
- 3.6. Recurrence relations and generating functions 113113
- 3.7. Rational generating functions and linear recursions 117117
- 3.8. Chromatic polynomials 120120
- 3.9. Combinatorial reciprocity 127127
- Exercises 130130

- Chapter 4. Counting with Exponential Generating Functions 138138
- Chapter 5. Counting with Partially Ordered Sets 160160
- 5.1. Basic properties of partially ordered sets 160160
- 5.2. Chains, antichains, and operations on posets 166166
- 5.3. Lattices 169169
- 5.4. The Möbius function of a poset 175175
- 5.5. The Möbius Inversion Theorem 178178
- 5.6. Characteristic polynomials 185185
- 5.7. Quotients of posets 189189
- 5.8. Computing the Möbius function 195195
- 5.9. Binomial posets 199199
- Exercises 204204

- Chapter 6. Counting with Group Actions 210210
- Chapter 7. Counting with Symmetric Functions 240240
- 7.1. The algebra of symmetric functions, Sym 240240
- 7.2. The Schur basis of Sym 245245
- 7.3. Hooklengths 251251
- 7.4. 𝑃-partitions 256256
- 7.5. The Robinson–Schensted–Knuth correspondence 261261
- 7.6. Longest increasing and decreasing subsequences 265265
- 7.7. Differential posets 269269
- 7.8. The chromatic symmetric function 274274
- 7.9. Cyclic sieving redux 277277
- Exercises 280280

- Chapter 8. Counting with Quasisymmetric Functions 288288
- Appendix. Introduction to Representation Theory 308308
- Bibliography 314314
- Index 318318