Permutations can be written as the "product" of "cycles", as described here [1].
If you choose an element belonging to a given cycle, then generate new elements by repeatedly applying the permutation, all the elements you generate will belong only to that cycle.
So, if your permutation can be written as the product of more than one cycle, and you choose a random starting element and generate new elements by successive application of the permutation, you will only generate a strict subset of all the possible elements.
On the other hand, if your permutation only has a single cycle, then that procedure will generate all the elements.
Hmm. I think my fundamental disconnect here was the idea of permutation as a repeated process. As a formal term it's really ever come up in a discrete mathematics class comparing it with combinations, on the subject of counting. Here we seem to be defining permutation as a function that can be repeatedly applied.
Looks like the first sentence of wikipedia confirms it's an overloaded word:
>In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set.
If you choose an element belonging to a given cycle, then generate new elements by repeatedly applying the permutation, all the elements you generate will belong only to that cycle.
So, if your permutation can be written as the product of more than one cycle, and you choose a random starting element and generate new elements by successive application of the permutation, you will only generate a strict subset of all the possible elements.
On the other hand, if your permutation only has a single cycle, then that procedure will generate all the elements.
[1] https://en.wikipedia.org/wiki/Permutation#Cycle_notation