Partitioning a list

[Oscar Benjamin]

On Thu, 23 Aug 2018 at 15:32, Sibylle Koczian <nulla.epistola at web.de> wrote:
>
> Am 21.08.2018 um 23:36 schrieb Poul Riis:
> > I would like to list all possible ways to put N students in groups of k students (suppose that k divides N) with the restriction that no two students should ever meet each other in more than one group.
> > I think this is a classical problem and I think there must be a python solution out there but I cannot find it. For instance, numpy's array_split only lists one (trivial) split.
> > I would be happy if someone could refer me to a general python algorithm solving the problem.
> >
> This is indeed a classic (or rather the generalization of a classic):
> Kirkman's Schoolgirl Problem (for N = 15), first published 1850.

I think the general concept is called a Steiner system:
https://en.wikipedia.org/wiki/Steiner_system

In the notation of that article you would like to describe S(2, k, N).
The article says:

    It can be shown that if there is a Steiner system S(2, k, N),
    where k is a prime power greater than 1, then N = 1 or k (mod k(k−1))

which may be relevant for what you are doing. For the case k=3 you can
find more information by searching for Steiner Triple Systems.

I don't know how to construct Steiner systems but without thinking
very hard about it I imagine that you can build larger Steiner systems
from smaller ones in some way. Once you have a Steiner system you can
then consider all the permutations of mapping your students onto the
elements of the system - there will be N! of those but possibly with a
k! duplication so maybe there's an efficient way to only enumerate the
N!/k! possibilities. Hopefully N isn't too large...

--
Oscar