§ Combinations notation in bijective combinatorics

They explicitly write nCrnCr as [n]C[r,nr][n]C[r, n-r]. This makes it better for "future uses", where it explicitly allows us to think of [n]C[x,y][n]C[x, y] as breaking nn into xx things we choose and yy things we don't choose. This makes the recurrence:
[n]C[r]=[n1]C[r1]+[n1]C[r] [n]C[r] = [n-1]C[r-1] + [n-1]C[r]
look as:
[n]C[r,nr]=[n1]C[r1,nr]+[n1]C[r,nr1] [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 ( r1r-1) or on the second component ( nr1n-r-1), in the smaller set ( n1n-1).