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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.

MSC:
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