In this paper, the following notation is defined:
For a vector $p$, we let $S_j(p)$ be the set of $j$ vertices $u$ maximizing $\frac{p(u)}{d(u)}$, breaking ties lexicographically. That is, $S_j(p) = \{ \pi(1), ..., \pi(j)\}$, where $\pi$ is the permutation such that $$\frac{p(\pi(i))}{d(\pi(i))} \ge \frac{p(\pi(i+1))}{d(\pi(i+1))}$$ for all $i$, and $\pi(i) \lt \pi(i+1)$ when those two ratios are equal.
($d(u)$ is the degree of vertex $u$)
What does permutation mean in this context?
As I see it, it is just an implication that the elements of $S_j(p)$ are sorted in the descending order of $\frac{p(i)}{d(i)}$.
EDIT: Thanks to joriki, I realize that the question might benefit from further explanation. I'm aware of the general definition of a permutation, as wikipedia says, "... a permutation of a set S is defined as a bijection from S to itself. That is, it is a function from S to S for which every element occurs exactly once as an image value." The issue here is that I'm confused by the particular wording of the excerpt and I'm not sure I'm interpreting it correctly, so the question's focus is not what a permutation is (in general), but rather how is it applied in this instance.