Notation about a randomized max cut algorithm.

http://users.cms.caltech.edu/~mccoy/publications/teatalk1.pdf

I'm trying to understand the lemma in this.

So we have

Lemma

Let $r$ be a random vector. For any unit vectors $u_{i}$ and $u_{j}$,

$\mathbb{P}\left(\operatorname{sign}(\langle u_{i},r\rangle) \neq \operatorname{sign}(\langle u_{j},r\rangle)\right) = \dfrac{\arccos(\langle u_{i},u_{j}\rangle)}{\pi}$

I'm a bit confused on notation. So what does $\mathbb{P}(\operatorname{sign}(\langle u_{i},r\rangle)$ mean. Also, I'm a bit confused on what a sign of a permutation is. I was under the impression that it's determined by the angle between the vectors.

You missed a parenthesis (which I have now fixed). It's actually $\mathbb{P}\left(\operatorname{sign}(\langle u_{i},r\rangle) \neq \operatorname{sign}(\langle u_{j},r\rangle)\right)$. Is that what your confusion was about?