What does the notation $ [ i \neq k ] $ mean?

Question

I can't figure out what the notation $[x \neq k ]$ means. Here's a bit of context:

The formula is: $Pr[A_i^k = 1] = \frac{[i\neq k]}{|k-i| + 1} = \begin{cases} \frac{1}{k-i+1} \text{ if } i \lt k \\ 0 \text { if } i = k \\ \frac{1}{i-k+1} \text{ if } i \gt k \end{cases}$

and is part of a chapter where the average expected time of operations of a randomised treap are proved.

$A_i^k$ is an indicator variable defined as $[ x_i \text{ is a proper ancestor of }x_k ]$ where $x_n$ is the node with the $n$-th smallest search key. That probability comes up because $\text{depth}(x_k) = \sum_{i=1}^{n} A_i^k$ and $\mathbf{E}[\text{depth}(x_k)] = \sum_{i=1}^nPr[A_i^k = 1]$.

I have no access to the pages that explain the notation since I'm studying from a pdf of a few pages taken from a book.

Answer 1

It is used like a boolean where $[i \neq k] = 1$ if $i \neq k$ and $0$ otherwise. Notice that $[i \neq k]$ is equivalent to $i < k$ or $i > k$ for numbers, which is the right part of the equation you wrote.

It is called Iverson bracket and in general $[statement] = 1$ if $statement$ is true and $0$ otherwise.