Stirling numbers are one of the most known classes of special numbers in Mathematics, especially in Combinatorics and Algebra. They were introduced by Scottish mathematician James Stirling (1692-1770) in his most important work, Differential Method with a Tract on Summation and Interpolation of Infinite Series (1730). Stirling numbers have a rich history; many arithmetic, number-theoretical, analytical and combinatorial connections; numerous classical properties; as well as many modern applications.This book collects much of the scattered material on the two subclasses of Stirling numbers to provide a holistic overview of the topic. From the combinatorial point of view, Stirling numbers of the second kind, S(n, k), count the number of ways to partition a set of n different objects (i.e., a given n-set) into k non-empty subsets. Stirling numbers of the first kind, s(n, k), give the number of permutations of n elements with k disjoint cycles. Both subclasses of Stirling numbers play an important role in Algebra: they form the coefficients, connecting well-known sets of polynomials.This book is suitable for students and professionals, providing a broad perspective of the theory of this class of special numbers, and many generalisations and relatives of Stirling numbers, including Bell numbers and Lah numbers. Throughout the book, readers are provided exercises to test and cement their understanding.

Combinatorics and Number Theory of Counting Sequences is an introduction to the theory of finite set partitions and to the enumeration of cycle decompositions of permutations. The presentation prioritizes elementary enumerative proofs. Therefore, parts of the book are designed so that even those high school students and teachers who are interested in combinatorics can have the benefit of them. Still, the book collects vast, up-to-date information for many counting sequences (especially, related to set partitions and permutations), so it is a must-have piece for those mathematicians who do research on enumerative combinatorics. In addition, the book contains number theoretical results on counting sequences of set partitions and permutations, so number theorists who would like to see nice applications of their area of interest in combinatorics will enjoy the book, too. Features The Outlook sections at the end of each chapter guide the reader towards topics not covered in the book, and many of the Outlook items point towards new research problems. An extensive bibliography and tables at the end make the book usable as a standard reference. Citations to results which were scattered in the literature now become easy, because huge parts of the book (especially in parts II and III) appear in book form for the first time.

This resource volume is an enlargement as well as an update of the previous edition. The book aims to introduce the reader to over 100 different families of positive integers. A brief historical note accompanies the descriptions and examples of several of the families together with a mix of routine exercises and problems as well as some thought provokers to solve. Number Treasury3 especially aims to stimulate further study beyond the scope of the introductory treatment given in the book. The emphasis in Number Treasury3 is on doing not proving. However, the reader is directed to think critically about situations, to provide explanations, to make generalizations, and to formulate conjectures. To engage the reader from the start, the book begins with a set of rich Investigations. These are standalone activities that represent each of the chapters of the book.

"...the great feature of the book is that anyone can read it without excessive head scratching...You'll find plenty here to keep you occupied, amused, and informed. Buy, dip in, wallow." -IAN STEWART, NEW SCIENTIST "...a delightful look at numbers and their roles in everything from language to flowers to the imagination." -SCIENCE NEWS "...a fun and fascinating tour of numerical topics and concepts. It will have readers contemplating ideas they might never have thought were understandable or even possible." -WISCONSIN BOOKWATCH "This popularization of number theory looks like another classic." -LIBRARY JOURNAL

Did you know there are 17 possible types of symmetric wallpaper pattern? Do you know what ‘casting out the nines’ is? Or why 88 is the fourth ‘untouchable’ number? Or how 7 is used to test for the onset of dementia. Number fanatic Derrick Niederman has a mission to bring numbers to life. He explores the unique properties of the most exciting numbers from 1 to 200, wherever they may crop up: from mathematics to sport, from history to the natural world, from language to pop culture. Packed with illustrations, amusing facts, puzzles, brainteasers and anecdotes, this is an enthralling and thought-provoking numerical voyage through the history of mathematics, investigating problems of logic, geometry and arithmetic along the way. ***PRAISE FOR THE REMARKABLE LIVES OF NUMBERS*** 'A hugely entertaining pick-and-mix of history, culture and mathematical puzzles.' BBC Focus 'This book is a complete joy. It made me smile. A lot.' Carol Vorderman 'Entertaining and engaging... Once you start reading it's just like the number system itself - impossible to stop.' Ian Stewart 'A fun book... definitely challenging.' Vanity Fair 'All sorts of fascinating mathematical minutiae.' Time Out

Figurate numbers have a rich history with many applications. The main purpose of this book is to provide a thorough and complete presentation of the theory of figurate numbers, giving much of their properties, facts and theorems with full proofs. This book is the first of this topic written in unified systematic way. It also contains many exercises with solutions.

Commutation Relations, Normal Ordering, and Stirling Numbers provides an introduction to the combinatorial aspects of normal ordering in the Weyl algebra and some of its close relatives. The Weyl algebra is the algebra generated by two letters U and V subject to the commutation relation UV - VU = I. It is a classical result that normal ordering pow

This volume, as Andrew M. Odlzyko writes in the foreword, “commemorates and celebrates the life and achievements of an extraordinary person.” Originally conceived as an 80th birthday tribute to Herbert Wilf, the well-known combinatorialist, the book has evolved beyond the proceeds of the W80 tribute. Professor Wilf was an award-winning teacher, who was supportive of women mathematicians, and who had an unusually high proportion of women among his PhD candidates. He was Editor-in-chief of the American Mathematical Monthly and a founder of both the Journal of Algorithms and of the Electronic Journal of Combinatorics. But he was first a researcher, driven by his desire to know and explain the inner workings of the mathematical world. The book collects high-quality, refereed research contributions by some of Professor Wilf’s colleagues, students, and collaborators. Many of the papers presented here were featured in the Third Waterloo Workshop on Computer Algebra (WWCA 2011, W80), held May 26-29, 2011 at Wilfrid Laurier University, Waterloo, Canada. Others were included because of their relationship to his important work in combinatorics. All are presented as a tribute to Herb Wilf’s contributions to mathematics and mathematical life.

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.