These are commonly known to mathematicians as integer partitions, thus the ways we can write an integer as a sum of smaller integers. But be careful, as the number of such ways quickly becomes huge for only reasonably small N. I posted somewhere a code that computes the actual number of ways you can do this. But Wolfram Alpha does it too. (There are something like 4 trillion distinct ways to write 200 as a sum of positive integers.)
I also posted the function partitions on the file exchange, that uses a recursive scheme to compute all integer partitions of any number. (Too large and you will be sorry of course.)
Each row is one such possibe partition. So the first row tells us that 3 = 1 + 1 + 1. In the last row, we only need one 3 to sum up to 3.
You can find partitions on the file exchange. Partitions has many options beyond the basic of course.