Combinatorics : The Rota Way - Joseph P. S. Kung

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Combinatorics

The Rota Way

By: Joseph P. S. Kung, Gian-Carlo Rota, Catherine H. Yan

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Paperback | 14 April 2009

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410 Pages

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22.86 x 15.24 x 2.31

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Gian-Carlo Rota was one of the most original and colourful mathematicians of the 20th century. His work on the foundations of combinatorics focused on the algebraic structures that lie behind diverse combinatorial areas, and created a new area of algebraic combinatorics. Written by two of his former students, this book is based on notes from his influential graduate courses and on face-to-face discussions. Topics include sets and valuations, partially ordered sets, distributive lattices, partitions and entropy, matching theory, free matrices, doubly stochastic matrices, Moebius 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. A must-have for all students and researchers in combinatorics and related areas.

Industry Reviews

'All in all ... I find Combinatorics: The Rota Way simply too good to pass up. Would that every day had 48 hours...' MAA Reviews
'... each of its chapters can be considered as a single expository work, conceived by Rota and masterfully moulded and improved by Kung and Yan.' Mathematical Reviews

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Preface	p. ix
Sets, Functions, and Relations	p. 1
Sets, Valuations, and Boolean Algebras	p. 1
Partially Ordered Sets	p. 9
Lattices	p. 17
Functions, Partitions, and Entropy	p. 28
Relations	p. 44
Further Reading	p. 52
Matching Theory	p. 53
What Is Matching Theory?	p. 53
The Marriage Theorem	p. 54
Free and Incidence Matrices	p. 62
Submodular Functions and Independent Matchings	p. 67
Rado's Theorem on Subrelations	p. 74
Doubly Stochastic Matrices	p. 78
The Gale-Ryser Theorem	p. 94
Matching Theory in Higher Dimensions	p. 101
Further Reading	p. 105
Partially Ordered Sets and Lattices	p. 106
Möbius Functions	p. 106
Chains and Antichains	p. 126
Sperner Theory	p. 136
Modular and Linear Lattices	p. 147
Finite Modular and Geometric Lattices	p. 161
Valuation Rings and Möbius Algebras	p. 171
Further Reading	p. 176
Generating Functions and the Umbral Calculus	p. 178
Generating Functions	p. 178
Elementary Umbral Calculus	p. 185
Polynomial Sequences of Binomial Type	p. 188
Sheffer Sequences	p. 205
Umbral Composition and Connection Matrices	p. 211
The Riemann Zeta Function	p. 218
Symmetric Functions and Baxter Algebras	p. 222
Symmetric Functions	p. 222
Distribution, Occupancy, and the Partition Lattice	p. 225
Enumeration Under a Group Action	p. 235
Baxter Operators	p. 242
Free Baxter Algebras	p. 246
Identities in Baxter Algebras	p. 253
Symmetric Functions Over Finite Fields	p. 259
Historical Remarks and Further Reading	p. 270
Determinants, Matrices, and Polynomials	p. 272
Polynomials	p. 272
Apolarity	p. 278
Grace's Theorem	p. 283
Multiplier Sequences	p. 291
Totally Positive Matrices	p. 296
Exterior Algebras and Compound Matrices	p. 303
Eigenvalues of Totally Positive Matrices	p. 311
Variation Decreasing Matrices	p. 314
Pólya Frequency Sequences	p. 317
Selected Solutions	p. 324
Bibliography	p. 369
Index	p. 389
Table of Contents provided by Ingram. All Rights Reserved.

Combinatorics

The Rota Way

At a Glance

Paperback

Industry Reviews

Shipping

How to return your order

Defective items

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