I am currently doing research in Combinatorial Geometry and I have been able to reduce a quite complicated problem relating to extending the Newton-Gregory problem of kissing spheres to a simple number theory problem and then checking every case to see that a conjecture of mine holds.
In any case, for background reasons which are not necessary for me to get into, I need to determine the explicit sets of 4 positive integers which when summed together give 12. Order does not matter as I will need to permute the set of 4 positive integers in each case to satisfy a different case for verifying my conjecture by exhaustion. So, I am not interested in some abstract "there are this many ways", I am actually interested in generating the explicit sets of numbers. I have been able to come up with the following so far:
$\{1,1,5,5\},\{2,2,4,4\},\{3,3,3,3\},\{2,2,3,5\},\{1,2,4,5\},\{1,3,3,5\},\{1,3,4,4\},\{2,3,3,4\}$
Any ideas for how to solve this? I can clarify any ambiguities as needed!
EDIT: I forgot to mention the following important detail:
I only want to consider integers from the set $\{2,3,4,5\}$ in summing to 12, since these correspond to the degree of a vertex and I have proved that for my particular problem that all vertices have either degree 2, 3, 4, or 5.