: Examines cyclic representations of permutations.

Perhaps the most diverse chapter in the book, Chapter 6 considers four interrelated topics: partitions, compositions, trees, and linear graphs. Partitions are ways of writing a number as a sum of positive integers, disregarding order; compositions consider order as important. Trees are connected acyclic graphs that arise in many contexts, from decision trees in computer science to phylogenetic trees in biology. Linear graphs (or paths) are among the simplest graph structures. The chapter connects these concepts to each other and demonstrates how generating functions can be used to enumerate them. An important feature of this chapter is the introduction of Pólya’s theory of counting in connection with counting trees, providing a glimpse into more advanced combinatorial methods.

The book provides an exhaustive analysis of how numbers and sets can be broken down into component parts.