GeneratingfunctionologyGeneratingfunctionology provides information pertinent to generating functions and some of their uses in discrete mathematics. This book presents the power of the method by giving a number of examples of problems that can be profitably thought about from the point of view of generating functions. Organized into five chapters, this book begins with an overview of the basic concepts of a generating function. This text then discusses the different kinds of series that are widely used as generating functions. Other chapters explain how to make much more precise estimates of the sizes of the coefficients of power series based on the analyticity of the function that is represented by the series. This book discusses as well the applications of the theory of generating functions to counting problems. The final chapter deals with the formal aspects of the theory of generating functions. This book is a valuable resource for mathematicians and students. |
Contents
| 1 | |
Chapter 2 Series | 27 |
The Exponential Formula | 64 |
Chapter 4 Applications of Generating Functions | 98 |
Chapter 5 Analytic and Asymptotic Methods | 138 |
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Common terms and phrases
analytic answer asymptotic average number Bell numbers binomial coefficient binomial type bipartite graph cards closed form combinatorial computation connected graphs converges counting defined disk divisors exactly example explicit formula exponential family exponential formula exponential generating function Fibonacci Fibonacci numbers formal power series free variable function f(z hand enumerator hands of weight Hence identity label sets labeled graphs left side legal string Lemma Let f(n letters log concave nonnegative integers number of hands number of objects number of partitions number of permutations opsgf ordinary power series pair parentheses partial fraction poles polynomial positive integers PP(f prefab problem proof properties prove radius of convergence recurrence relation relabeling result rooted trees series ƒ series generating function sieve method singularities Snake Oil method solve Stirling numbers subset summation Suppose theorem unimodal v₁ values vertex vertices zeros