Enumerative combinatorics has undergone enormous development since the publication of the first edition of this book in 1986. It has become more clear what are the essential topics, and many interesting new ancillary results have been discovered. This second edition is an attempt to bring the coverage of the first volume more up-to-date and to impart a wide variety of additional applications and examples.

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This is the second (and final) volume of a graduate-level introduction to enumerative combinatorics. To those who have been waiting twelve years since the publication of Volume 1, I can only say that no one is more pleased to see Volume 2 finally completed than myself. I have tried to cover what I feel are the fundamental topics in enumerative combinatorics, and the ones that are the most useful in applications outside of combinatorics. Though the book is primarily intended to be a textbook for graduate students and a resource for professional mathematicians, I hope that undergraduates and even bright high-school students will find something of interest. For instance, many of the 66 combinatorial interpretations of Catalan numbers provided by Exercise 6.19 should be accessible to undergraduates with a little knowledge of combinatorics.

Revising this textbook has been a special challenge, for a very nice reason. So many people have read this book, and taught from it, and even loved it. The spirit of the book could never change. This text was written to help our teaching of linear algebra keep up with the enormous importance of this subject—which just continues to grow.

Walter rudin is the author of threee textbooks, Principles of Mathematical Analysis, Real and Complex Analysis, and Functional Analysis

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Numerical mathematics is the branch of mathematics that proposes, develops, analyzes and applies methods from scientific computing to several fields including analysis, linear algebra, geometry, approximation theory, functional equations, optimization and differential equations. Other disciplines such as physics, the natural and biological sciences, engineering, and economics and the financial sciences frequently give rise to problems that need scientific computing for their solutions.

We have developed this material for a sequence of courses on the theory and application of numerical approximation techniques. The text is designed primarily for junior-level mathematics, science, and engineering majors who have completed at least the first year of the standard college calculus sequence. Familiarity with the fundamentals of matrix algebra and differential equations is also useful, but adequate introductory material on these topics is presented in the text so that those courses need not be prerequisites.

A First Course in Abstract Algebra introduces groups and commutative rings. Group theory was invented by E. Galois in the early 1800s, when he used groups to completely determine when the roots of polynomials can be found by formulas generalizing the quadratic formula.