Analytic combinatorics uses techniques from complex analysis to solve problems in enumerative combinatorics, specifically to find asymptotic estimates for the coefficients of generating functions.Melczer 2021, pp. vii and ix.Pemantle and Wilson 2013, pp. xi.Flajolet and Sedgewick 2009, pp. ix.
History
One of the earliest uses of analytic techniques for an enumeration problem came from Srinivasa Ramanujan and G. H. Hardy's work on integer partitions,Melczer 2021, pp. vii.Pemantle and Wilson 2013, pp. 62-63. starting in 1918, first using a Tauberian theorem and later the circle method.Pemantle and Wilson 2013, pp. 62.
Walter Hayman's 1956 paper "A Generalisation of Stirling's Formula" is considered one of the earliest examples of the saddle-point method.Pemantle and Wilson 2013, pp. 63.Wilf 2006, pp. 197.Flajolet and Sedgewick 2009, pp. 607.
In 1990, Philippe Flajolet and Andrew Odlyzko developed the theory of singularity analysis.Flajolet and Sedgewick 2009, pp. 438.
In 2009, Philippe Flajolet and Robert Sedgewick wrote the book Analytic Combinatorics, which presents analytic combinatorics with their viewpoint and notation.
Some of the earliest work on multivariate generating functions started in the 1970s using probabilistic methods.Melczer 2021, pp. 13.Flajolet and Sedgewick 2009, pp. 650 and 717.
Development of further multivariate techniques started in the early 2000s.Melczer 2021, pp. 13-14.
Techniques
Meromorphic functions
If is a meromorphic function and is its pole closest to the origin with order , thenSedgewick 4, pp. 59 as where is the coefficient of in the Taylor expansion of around .
Tauberian theorem
If as where and is a slowly varying function, thenFlajolet and Sedgewick 2009, pp. 435. Hardy 1949, pp. 166. I use the form in which it is stated by Flajolet and Sedgewick. as
See also the Hardy–Littlewood Tauberian theorem.
Circle Method
For generating functions with logarithms or roots, which have branch singularities.Pemantle and Wilson 2013, pp. 55-56.
Darboux's method
If we have a function where and has a radius of convergence greater than and a Taylor expansion near 1 of , thenWilf 2006, pp. 194.
See Szegő (1975) for a similar theorem dealing with multiple singularities.
Singularity analysis
If has a singularity at and
as
where thenFlajolet and Sedgewick 2009, pp. 393.
as
Saddle-point method
For generating functions including entire functions.Wilf 2006, pp. 196.Flajolet and Sedgewick 2009, pp. 542.
Intuitively, the biggest contribution to the contour integral is around the saddle point and estimating near the saddle-point gives us an estimate for the whole contour.
If is an admissible function,See Flajolet and Sedgewick 2009, pp. 565 or Wilf 2006, pp. 199. thenFlajolet and Sedgewick 2009, pp. 553.Sedgewick 8, pp. 25.
as
where .
See also the method of steepest descent.
Notes
References
Further reading
External links Analytic Combinatorics online course An Introduction to the Analysis of Algorithms online course Analytic Combinatorics in Several Variables projects An Invitation to Analytic Combinatorics
See also Symbolic method (combinatorics) Analytic Combinatorics (book)