Efficiently computing vector combinations

Question

I'm working on a research problem out of curiosity, and I don't know how to program the logic that I've in mind. Let me explain it to you:

I've four vectors, say for example,

v1 = 1 1 1 1
v2 = 2 2 2 2
v3 = 3 3 3 3
v4 = 4 4 4 4

Now what I want to do is to add them combination-wise, that is,

v12 = v1+v2
v13 = v1+v3
v14 = v1+v4
v23 = v2+v3
v24 = v2+v4
v34 = v3+v4

Till this step it is just fine. The problem is now I want to add each of these vectors one vector from v1, v2, v3, v4 which it hasn't added before. For example:

v3 and v4 hasn't been added to v12, so I want to create v123 and v124. Similarly for all the vectors like,

v12 should become:
v123 = v12+v3
v124 = v12+v4

v13 should become:
v132 // This should not occur because I already have v123
v134

v14 should become:
v142 // Cannot occur because I've v124 already
v143 // Cannot occur

v23 should become:
v231 // Cannot occur
v234 ... and so on.

It is important that I do not do all at one step at the start. Like for example, I can do (4 choose 3) 4C3 and finish it off, but I want to do it step by step at each iteration.

How do I program this?

P.S.: I'm trying to work on an modified version of an apriori algorithm in data mining.