Combinatorics: The Rota Way

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Written by two of his former students, this book is based on notes from his courses and on personal discussions with him. Topics include sets and valuations, partially ordered sets, distributive lattices, partitions and entropy, matching theory, free matrices, doubly stochastic matrices, M&oumlet;bius functions, chains and antichains, Sperner theory, commuting equivalence relations and linear lattices, modular and geometric lattices, valuation rings, generating functions, umbral calculus, symmetric functions, Baxter algebras, unimodality of sequences, and location of zeros of polynomials. Many exercises and research problems are included and unexplored areas of possible research are discussed.
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Editorial Reviews

From the Publisher
"...simply too good to pass up."
Michael Berg, MAA Reviews

"...stimulating, valuable book..."
D.V. Chopra, Choice Magazine

"Students and researchers of combinatorics, as well as logicians and computer scientists, should keep a copy on their bookshelves."
Walter Carnielli,

"A mathematician or computer scientist wanting to learn more about the science of combinatorics will find a good read here. The content is well written, very accurate and well edited."
John Mount, SIGACT News

"This book represents a valuable contribution to combinatorics in general, and I think that it could be useful to all researchers dealing with matters of this kind, especially younger ones, who intend to spend much of their future efforts on problems of a combinatorial nature."
Luca Ferrari, Mathematical Reviews

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Product Details

  • ISBN-13: 9780521737944
  • Publisher: Cambridge University Press
  • Publication date: 2/28/2009
  • Series: Cambridge Mathematical Library Series
  • Edition description: New Edition
  • Pages: 408
  • Product dimensions: 6.00 (w) x 8.90 (h) x 1.00 (d)

Meet the Author

Dr Joseph P. S. Kung is Professor of Mathematics at the University of North Texas. He is currently an editor-in-chief for the journal Advances in Applied Mathematics.

Gian-Carlo Rota (1932–1999) was a Professor of Applied Mathematics and Natural Philosophy at the Massachusetts Institute of Technology. He was a member of the National Academy of Science. He was awarded the 1988 Steele Prize of the American Mathematical Society for his 1964 paper 'On the Foundations of Combinatorial Theory I. Theory of Moebius Functions.' He was the founding editor of the Journal of Combinatorial Theory.

Catherine H. Yan is a Professor of Mathematics at Texas A & M University. She was a Courant Instructor at New York University (1997–1999) and a Sloan Fellow (2001–2003).

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Table of Contents

Preface ix

1 Sets, Functions, and Relations 1

1.1 Sets, Valuations, and Boolean Algebras 1

1.2 Partially Ordered Sets 9

1.3 Lattices 17

1.4 Functions, Partitions, and Entropy 28

1.5 Relations 44

1.6 Further Reading 52

2 Matching Theory 53

2.1 What Is Matching Theory? 53

2.2 The Marriage Theorem 54

2.3 Free and Incidence Matrices 62

2.4 Submodular Functions and Independent Matchings 67

2.5 Rado's Theorem on Subrelations 74

2.6 Doubly Stochastic Matrices 78

2.7 The Gale-Ryser Theorem 94

2.8 Matching Theory in Higher Dimensions 101

2.9 Further Reading 105

3 Partially Ordered Sets and Lattices 106

3.1 M&oumlet;bius Functions 106

3.2 Chains and Antichains 126

3.3 Sperner Theory 136

3.4 Modular and Linear Lattices 147

3.5 Finite Modular and Geometric Lattices 161

3.6 Valuation Rings and M&oumlet;bius Algebras 171

3.7 Further Reading 176

4 Generating Functions and the Umbral Calculus 178

4.1 Generating Functions 178

4.2 Elementary Umbral Calculus 185

4.3 Polynomial Sequences of Binomial Type 188

4.4 Sheffer Sequences 205

4.5 Umbral Composition and Connection Matrices 211

4.6 The Riemann Zeta Function 218

5 Symmetric Functions and Baxter Algebras 222

5.1 Symmetric Functions 222

5.2 Distribution, Occupancy, and the Partition Lattice 225

5.3 Enumeration Under a Group Action 235

5.4 Baxter Operators 242

5.5 Free Baxter Algebras 246

5.6 Identities in Baxter Algebras 253

5.7 Symmetric Functions Over Finite Fields 259

5.8 Historical Remarks and Further Reading 270

6 Determinants, Matrices, and Polynomials 272

6.1 Polynomials 272

6.2 Apolarity 278

6.3 Grace's Theorem 283

6.4 Multiplier Sequences 291

6.5 TotallyPositive Matrices 296

6.6 Exterior Algebras and Compound Matrices 303

6.7 Eigenvalues of Totally Positive Matrices 311

6.8 Variation Decreasing Matrices 314

6.9 Pólya Frequency Sequences 317

Selected Solutions 324

Bibliography 369

Index 389

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