Ordered Genes In this chapter we study genes that are ordered lists of objects without repetition. These Summary: Chapter 4 Ordered Genes In this chapter we study genes that are ordered lists of objects without repetition. These are useful for classical optimization problems like the famous traveling salesman problem as well as for evolving controls on a greedy algorithm. There is a good deal of mathematical structure that applied to ordered genes, viewed as permutations. A useful subset of this lore appears at the end of the chapter in Section 4.3. We will begin by looking at possible variation operators that operate directly on an ordered list. In later sections we will look at alternate representations. 4.1 Variation Operators There is a very natural choice for unary variation operators for ordered lists: the transposi- tion. Definition 4.1 A transpositional mutation exchanges two elements of an ordered list. These elements are selected uniformly at random. Theorem 4.2, in the mathematical background section, shows that it is possible to trans- form any ordered gene into any other by a sequence of transpositional mutations. This is a property that is taken for granted in most evolutionary computation systems, but which sometimes causes problems. For this reason, we give the property a name. Definition 4.2 A set of mutation operators is complete if a finite number of mutations can transform any potential population member into any other. Collections: Mathematics