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Asymptotic enumeration methods. (English) Zbl 0845.05005
Graham, R. L. (ed.) et al., Handbook of combinatorics. Vol. 1-2. Amsterdam: Elsevier (North-Holland). 1063-1229 (1995).
This magnificent article from the “Handbook of combinatorics” is a must for all mathematicians and computer scientists who are interested in the asymptotic aspects of enumerative sequences. The author, who is one of the leading experts in the field, gives a detailed overview on the analytical toolkit as well as a huge variety of examples. Amongst the themes in consideration there are direct asymptotic methods like Euler-Mclaurin and Poisson summation, as well as methods based on generating functions like singularity analysis (transfer theorems, Darboux’s method, saddle point method, circle method) or Mellin transforms. A final section on the very recent developments in algorithmic and automated asymptotics (initialized by P. Flajolet and his group at INRIA) and a long list of references finish this impressive overview of an area on the borderline of analysis and discrete mathematics.

05A16Asymptotic enumeration
05A15Exact enumeration problems, generating functions
05A10Combinatorial functions
30B10Power series (one complex variable)
68Q25Analysis of algorithms and problem complexity
68R05Combinatorics in connection with computer science