Combinatorics The Art Of Counting

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.

This book provides an introduction to discrete mathematics. At the end of the book the reader should be able to answer counting questions such as: How many ways are there to stack n poker chips, each of which can be red, white, blue, or green, such that each red chip is adjacent to at least 1 green chip? The book can be used as a textbook for a semester course at the sophomore level. The first five chapters can also serve as a basis for a graduate course for in-service teachers.

Written by two well-known scholars in the field,Combinatorial Reasoning: An Introduction to the Art ofCounting presents a clear and comprehensive introduction to theconcepts and methodology of beginning combinatorics. Focusing onmodern techniques and applications, the book develops a variety ofeffective approaches to solving counting problems. Balancing abstract ideas with specific topical coverage, thebook utilizes real world examples with problems ranging from basiccalculations that are designed to develop fundamental concepts tomore challenging exercises that allow for a deeper exploration ofcomplex combinatorial situations. Simple cases are treated firstbefore moving on to general and more advanced cases. Additionalfeatures of the book include: • Approximately 700 carefully structured problems designedfor readers at multiple levels, many with hints and/or shortanswers • Numerous examples that illustrate problem solving usingboth combinatorial reasoning and sophisticated algorithmicmethods • A novel approach to the study of recurrence sequences,which simplifies many proofs and calculations • Concrete examples and diagrams interspersed throughout tofurther aid comprehension of abstract concepts • A chapter-by-chapter review to clarify the most crucialconcepts covered Combinatorial Reasoning: An Introduction to the Art ofCounting is an excellent textbook for upper-undergraduate andbeginning graduate-level courses on introductory combinatorics anddiscrete mathematics.

Mathematics is the science of patterns, and mathematicians attempt to understand these patterns and discover new ones using a variety of tools. In Proofs That Really Count, award-winning math professors Arthur Benjamin and Jennifer Quinn demonstrate that many number patterns, even very complex ones, can be understood by simple counting arguments. The book emphasizes numbers that are often not thought of as numbers that count: Fibonacci Numbers, Lucas Numbers, Continued Fractions, and Harmonic Numbers, to name a few. Numerous hints and references are given for all chapter exercises and many chapters end with a list of identities in need of combinatorial proof. The extensive appendix of identities will be a valuable resource. This book should appeal to readers of all levels, from high school math students to professional mathematicians.

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 abstra

In this book the authors aim to endow the reader with an operational, conceptual, and methodological understanding of the discrete mathematics that can be used to study, understand, and perform computing. They want the reader to understand the elements of computing, rather than just know them. The basic topics are presented in a way that encourages readers to develop their personal way of thinking about mathematics. Many topics are developed at several levels, in a single voice, with sample applications from within the world of computing. Extensive historical and cultural asides emphasize the human side of mathematics and mathematicians. By means of lessons and exercises on “doing” mathematics, the book prepares interested readers to develop new concepts and invent new techniques and technologies that will enhance all aspects of computing. The book will be of value to students, scientists, and engineers engaged in the design and use of computing systems, and to scholars and practitioners beyond these technical fields who want to learn and apply novel computational ideas.