Stirling Numbers

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.

Contents:

• Preface
• About the Author
• Notations
• Preliminaries
• Combinatorics of Partitions
• Stirling Numbers of the Second Kind
• Stirling Numbers of the First Kind
• Generalisations and Relatives of Stirling Numbers
• Zoo of Numbers
• Mini Dictionary
• Exercises
• Bibliography
• Index

Readership: Teachers and students (esp. at university) interested in Combinatorics, Number Theory, General Algebra, Cryptography and related fields, as well as general audience of amateurs of Mathematics.
Key Features:

• Books in this series provide a complete presentation of the Theory of two classes of special numbers (Stirling numbers of the first and of the second kind) and to give much of their properties, facts and theorems with full proofs of them
• Collects much scattered material and presents updated material with all details in clear and unified way
• Consider all ranges of well-known and hidden connections of a given set number with different mathematical problems; draw up a system of multilevel tasks
• Collect and study a large list of generalizations and relatives of Stirling numbers (Lah numbers, Bell numbers, factorial numbers, etc.)