Analytic Combinatorics — Symbolic Combinatorics

This booklet develops in nearly 200 pages the basics of combinatorial enumeration through an approach that revolves around generating functions. The major objects of interest here are words, trees, graphs, and permutations, which surface recurrently in all areas of discrete mathematics. The text presents the core of the theory with chapters on unlabelled enumeration and ordinary generating functions, labelled enumeration and exponential generating functions, and finally multivariate enumeration and generating functions. It is largely oriented towards applications of combinatorial enumeration to random discrete structures and discrete mathematics models, as they appear in various branches of science, like statistical physics, computational biology, probability theory, and, last not least, computer science and the analysis of algorithms. Acknowledgements. This work was supported in part by the IST Programme of the EU under contract number IST-1999-14186 (ALCOM-FT). The authors are grateful to Xavier Gourdon who incited us to add a separate chapter on multivariate generating functions and to Brigitte Vallée for many critical suggestions regarding the presentation and global organization of this text. This booklet would be substantially different (and much less informative) without Neil Sloane’s Encyclopedia of Integer Sequences, Steve Finch’s Mathematical Constants, both available on the internet. Bruno Salvy and Paul Zimmermann have developed algorithms and libraries for combinatorial structures and generating functions that are based on the MAPLE system for symbolic computations and have proven to be immensely useful. “Symbolic Combinatorics” is a set of lecture notes that are a component of a wider book project titled Analytic Combinatorics, which will provide a unified treatment of analytic methods in combinatorics. This text is partly based on an earlier document titled “The Average Case Analysis of Algorithms: Counting and Generating Functions”, INRIA Res. Rep. #1888 (1993), 116 pages, which it now subsumes. c Philippe Flajolet and Robet Sedgewick, May 2002.