Élie de Panafieu

This book will transform the way you analyze sequences. Before reading it, my goto technique was to write down a recurrence and wrestle with it. My first thought about introducing a series with my sequence as coefficients was that this could only make the problem worst. If I know nothing about my sequence beside a recurrence, I certainly know even less about this strange so-called 'generating function'! This book showed me how wrong my first impression was. Generating functions simplify the analysis of sequences in many ways and are now my favorite technique. The book is driven by application, in particular to the analysis of algorithms. Just to cite a few at the top of my mind - How many people should be gathered before two have the same birthday? (around 24) - How many times would you expect to draw in an urn with replacement before having seen each ball it contains? - How many words of length n belong to a given rational language? Same question for an algebraic language? - Counting connected graphs - How many ways are there to write a large integer as a sum of primes? - What is the probability that a random large permutation has no fixed point? (around 1/e) Generating functions can be thought of like nice compact representations of recurrences. But as shown by the book, they are more than that. First, it presents the 'symbolic method' to translate the combinatorial structure of the objects enumerated directly into generating function relations, without the need to go through recurrence. Then, the book introduces techniques to extract from the generating functions asymptotics and various statistics of the sequence. Those techniques rely on complex analysis and are often inspired by / borrowed from analytic arithmetic. But they are presented from scratch and very clearly, so even someone without any complex analysis background can easily follow (knowing about series helps).

Clear advices to improve code readability and structure. A reference in the field. Java examples are provided. I would have appreciate a bit more about class definition and more difficult cases for the single responsibility principle.