Problem 9:

Given a natural number n, list all the ways that it can written as a sum of smaller (or equal) natural numbers.  Such a sum is called  a partition of n. And since order of summation is irrelevant, we may assume that the summands are written in increasing order.  For example for n=5, we get the following list of all 7 partitions:

5=1+1+1+1+1    =                

5=1+1+1+2        =                

5=1+1+3             =                

5=1+2+2             =                

5=1+4                 =                

5=2+3                 =                

5=5                      =                

The right column gives a way of abbreviating partitions.  Using this abbreviation, I can finally state the equation that you are to prove for this problem.  Here it is:  prove that for any n,

Sticking with our earlier example, what this says for n=5 is: