Combinatorics: The Art of Counting (Graduate Studies in Mathematics)

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. 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 q-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. P-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 P-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 "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 mathmatics in general and combinatorics in particular."--p.4 ppr.wrppr Offers a gentle introduction to the enumerative part of combinatorics. In addition to covering all the standard techniques for counting combinatorial objects, the text contains material from the research literature such as the use of quotient posets to study the Moebius function and characteristic polynomial of a partially ordered set. "The pdf contains a draft title page, draft copyright page and a draft manuscript"-- Provided by publisher