Analytic Combinatorics

Philippe Flajolet and Robert Sedgewick, "Analytic Combinatorics"
Web Edition. Ninth Printing (Valentine's) | February 14, 2007 | no ISBN | PDF | 760 pages | 10,9 M
Analytic Combinatorics aims at predicting precisely the properties of large structured combinatorial configurations, through an approach based extensively on analytic methods. Generating functions are the central objects of the theory.

Analytic Combinatorics starts from an exact enumerative description of combinatorial structures by means of generating functions, which make their first appearance as purely formal algebraic objects. Next, generating functions are interpreted as analytic objects, that is, as mappings of the complex plane into itself. Singularities determine a function��s coefficients in asymptotic form and lead to precise estimates for counting sequences. This chain applies to a large number of problems of discrete mathematics relative to words, trees, permutations, graphs, and so on. A suitable adaptation of the methods also opens the way to the quantitative analysis of characteristic parameters of large random structures, via a perturbational approach.

This book is meant to be reader-friendly. Each major method is abundantly illustrated by means of concrete examples treated in detail -- there are scores of them, spanning froma fraction of a page to several pages -- offering a complete treatment of a specific problem. These are borrowed not only from combinatorics itself but also from neighbouring areas of science. With a view of addressing not only mathematicians of varied profiles but also scientists of other disciplines, Analytic Combinatorics is self contained, including ample appendices that recapitulate the necessary background in combinatorics and complex function theory. A rich set of short Notes -- there are more than 250 of them -- are inserted in the text and can provide exercises meant for self study or for students' practice, as well as introductions to the vast body of literature that is available. We have also made every effort to focus on core ideas rather than technical details, supposing a certain amount of mathematical maturity but only basic prerequisites on the part of our gentle readers. The book is also meant to be strongly problem-oriented, and indeed it can be regarded as amanual, or even a huge algorithm, guiding the reader to the solution of a very large variety of problems regarding discrete mathematical models of varied origins. In this spirit, many of our developments connect nicely with computer algebra and symbolic manipulation systems.

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