Question

I have n elements that need to be partitioned into x sets, each set has to hold exactly k=4 elements.

I need to find all possible partitions with the constraint that each pair of elements only shares the same set once.

So if I start with [1 2 3 4] [5 6 7 8] […], all consecutive partitions cannot hold e.g. [1 2 X X] or [X X 1 3]. sets are unordered.

Close to this problem are the stirling numbers of the second kind. However, they only solve the problem for arbitrarily sized sets.

Example: I have 32 mice that can be put in 8 cages, 4 per cage. The mice should be rotated between the cages in a fashion that they never meet another mouse twice. How often can you do this and what are the configurations?

Answer 1

This is an instance of the “social golfer problem.” Warwick Harvey used to have a page (http://www.cs.st-andrews.ac.uk/~wh/golf/) with a bunch of solutions for different problem sizes, but it seems to be down. The answer in your case turns out to be 10 rotations, but I don’t know what the actual configurations are. Here is a 9-rotation solution, though: http://www.cs.st-andrews.ac.uk/~ianm/CSPLib//prob/prob010/solution

It is an unsolved problem for general n and k.