4.   Complex Analysis, Rational and Meromorphic Asymptotic



IV.1 Generating functions as analytic objects


IV.2 Analytic functions and meromorphic functions


IV.3 Singularities and exponential growth of coefficients


IV.4 Closure properties and computable bounds


IV.5 Rational and meromorphic functions


IV.6 Localization of singularities


IV.7 Singularities and functional equations


IV.8 Perspective


Selected Exercises

Note IV.28

Supernecklaces. A "supernecklace" of the 3rd type is a labelled cycle of cycles (see page 125). Draw all the supernecklaces of the 3rd type of size $n$ for $n$ = 1, 2, 3, and 4. Then develop an asymptotic estimate of the number of supernecklaces of size $n$ by showing that $$[z^n]\ln\Bigl({1\over 1 - \ln{1\over 1-z}}\Bigr)\sim{1\over n}(1-e^{-1})^{-n}.$$ Hint: Take derivatives.



Selected Experiments

Program IV.1

Compute the percentage of permutations having no singelton or doubleton cycles and compare with the asymptotic estimate from analytic combinatorics, for N = 10 and N = 20 .


Program IV.2

Plot the derivative of the supernecklace GF (see Note IV.28) in the style of the plots in Lecture 4. Click here for access to the Java code in the lecture.