Combinatorics The Rota Way 1st Edition by Joseph Kung, Gian Carlo Rota, Catherine Yan 9780511501111 0511501110

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Product details:

ISBN 10: 0511501110
ISBN 13: 9780511501111
Author: Joseph Kung, Gian Carlo Rota, Catherine Yan

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.

Combinatorics The Rota Way 1st Table of contents:

1 Sets, Functions, and Relations

1.1 Sets, Valuations, and Boolean Algebras

Exercises

1.2 Partially Ordered Sets

Exercises

1.3 Lattices

Exercises

1.4 Functions, Partitions, and Entropy

Exercises

1.5 Relations

Exercises

1.6 Further Reading

2 Matching Theory

2.1 What Is Matching Theory?

Exercises

2.2 The Marriage Theorem

Exercises

2.3 Free and Incidence Matrices

Exercises

2.4 Submodular Functions and Independent Matchings

Exercises

2.5 Rado’s Theorem on Subrelations

Exercises

2.6 Doubly Stochastic Matrices

Exercises

2.7 The Gale–Ryser Theorem

Exercises

2.8 Matching Theory in Higher Dimensions

Exercises

2.9 Further Reading

3 Partially Ordered Sets and Lattices

3.1 Möbius Functions

Exercises

3.2 Chains and Antichains

Exercises

3.3 Sperner Theory

Exercises

3.4 Modular and Linear Lattices

Exercises

3.5 Finite Modular and Geometric Lattices

Exercises

3.6 Valuation Rings and Möbius Algebras

Exercises

3.7 Further Reading

4 Generating Functions and the Umbral Calculus

4.1 Generating Functions

Exercises

4.2 Elementary Umbral Calculus

Exercises

4.3 Polynomial Sequences of Binomial Type

Exercises

4.4 Sheffer Sequences

Exercises

4.5 Umbral Composition and Connection Matrices

Exercises

4.6 The Riemann Zeta Function

Exercises

5 Symmetric Functions and Baxter Algebras

5.1 Symmetric Functions

Exercises

5.2 Distribution, Occupancy, and the Partition Lattice

Exercises

5.3 Enumeration Under a Group Action

Exercises

5.4 Baxter Operators

Exercises

5.5 Free Baxter Algebras

Exercises

5.6 Identities in Baxter Algebras

Exercises

5.7 Symmetric Functions Over Finite Fields

Exercises

5.8 Historical Remarks and Further Reading

6 Determinants, Matrices, and Polynomials

6.1 Polynomials

Exercises

6.2 Apolarity

Exercises

6.3 Grace’s Theorem

Exercises

6.4 Multiplier Sequences

Exercises

6.5 Totally Positive Matrices

Exercises

6.6 Exterior Algebras and Compound Matrices

Exercises

6.7 Eigenvalues of Totally Positive Matrices

6.8 Variation-Decreasing Matrices

Exercises

6.9 Pólya Frequency Sequences

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Combinatorics,Joseph Kung,Gian Carlo Rota,Catherine Yan