# 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.