§ Combinations notation in bijective combinatorics

They explicitly write $nCr$ as $[n]C[r, n-r]$. This makes it better for "future uses", where it explicitly allows us to think of $[n]C[x, y]$ as breaking $n$ into $x$ things we choose and $y$ things we don't choose. This makes the recurrence:

$$[n]C[r] = [n-1]C[r-1] + [n-1]C[r]$$

look as:

$$[n]C[r,n-r] = [n-1]C[r-1,n-r] + [n-1]C[r, n-r-1]$$

That is, we are reducing on either the first component ( $r-1$) or on the second component ( $n-r-1$), in the smaller set ( $n-1$).