## § 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 ($n-1$).