# Combinatorics II Problems and Solutions: Counting Patterns

Stefan Hollos and J. Richard Hollos

**Format and pricing:** Paperback (219 pages) $16.95, Kindle/pdf $9.95

**ISBN:** 9781887187329 (paperback), 9781887187336 (ebook)

**Publication date:** November 2016

## Table of Contents

- Preface
- Counting Words
- Regular Expressions
- Finite Automata
- Generating Functions
- Counting Necklaces
- Counting Bracelets
- Polya and Burnside Theorems
- Problems and Solutions
- Totient Function
- Matrix Inversion
- Software
- References and Further Reading
- Acknowledgements
- About the Authors
- Index

This book deals mainly with pattern counting problems. It is a continuation of our previous combinatorics problem book. There are 80 problems with detailed solutions, including 70 figures, many of which are examples of patterns.

The book will teach you powerful methods for counting patterns. These methods should be in the toolbox of every combinatorialist. It also provides the means to generate patterns with programs that can be downloaded below.

The book starts with patterns that can be described by regular expressions and finite automata. It shows how to get generating functions for families of patterns from a regular expression or it's corresponding finite automaton.

It then looks at pattern counting problems that involve equivalence under symmetry. For example, how many unique necklaces can one construct using beads of 3 different colors if a rotated necklace is considered the same as the original?

These problems are surprisingly easy to answer using a method called Polya's theory of counting. This method and its more general form, called Burnside's theorem are covered. There are many worked out problems that show how to use these methods. Included are problems that find the number of unique ways to color the Platonic solids.

#### About the authors

Stefan Hollos and J. Richard Hollos are physicists and electrical engineers by training, and enjoy anything related to math, physics, engineering and computing. They are brothers and business partners at Exstrom Laboratories LLC in Longmont, Colorado.

Send comments to: Richard Hollos (richard[AT]exstrom DOT com)

Copyright 2019 by Exstrom Laboratories LLC