# 5. Applications of Rational and Meromorphic Asymptotics

## V.1 A roadmap to rational and meromorphic asymptotics

## V.2 The supercritical sequence schema

## V.3 Regular specifications and languages

## V.4 Nested sequences, lattice paths, and continued fractions

## V.5 Paths in graphs and automata

## V.6 Transfer matrix models

## V.7 Perspective

#### Web Exercises

## V.1

Give an asymptotic expression for the number of strings that do not contain the pattern`0000000001`. Do the same for

`0101010101`.

## V.2

Give asymptotic expressions for the number of objects of size*N*and the number of parts in a random object of size

*N*for the following classes: compositions of 1s, 2s, and 3s, triple surjections, and alignments with no singleton cycles.

#### Selected Experiments

## Program V.1

In the style of the plots in the Lecture 6, plot the GFs for the set of bitstrings having no occurrence of the pattern`0000000001`. Do the same for

`0101010101`. (See Web Exercise V.1 and Program IV.2).