Combinatorics of Permutations, 3rd Edition

3In Many Circles. Permutations as Products of Cycles.

DOI: 10.1201/9780429274107-3

3.1 Decomposing a Permutation into Cycles

So far we have looked at permutations as linear orders, that is, ways of listing n objects so that each object gets listed exactly once. In this section we will discuss permutations from a different viewpoint. We will consider them as functions. Let us redefine permutations along these lines.

DEFINITION 3.1 Let $f:[n]\to [n]$ be a bijection. Then we say that f is a permutation of the set [n].

This definition certainly does not contradict our former definition of permutations. Formerly, we said that 34152 was a permutation of length five. Now we can reformulate that sentence by saying that the function $f:[5]\to [5]$ defined ...