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).