Try the following: every expression of gives us an expression for by adding it 2 to the left of each sum, and each expression of gives us a new expression for by adding 1 to the leftmost summand in each sum, when we order the sums in each case just as you did: in increasing order from left to right.
For example, for
Note that the sums between *.* are constructed from sums of 7 by adding 1 to each leftmost number there, whereas the other sums are constructed from the ones of 6 by adding 2 in the left to each sum there.
The only thing left there is to prove that every sum for is always gotten that way, but that is easy: if the sum begins with a 2 it comes from , otherwise it comes from