# set partitioning

Boris Borcic bborcic at gmail.com
Tue May 2 13:49:34 CEST 2006

```hymort at hotmail.com wrote:
> For the list [1,2,3,4], I'm looking for the following for k = 2:
>
> [[1,2], [3,4]]
> [[1,3], [2,4]]
> [[1,4], [2,3]]
>
> for k = 3:
>
> [[1,2,3], ]
> [[1,2,4], ]
> [[1,3,4], ]
> [[2,3,4], ]
>

def pickk(k,N,m=0) :
if k==1 :
return ((n,) for n in range(m,N))
else :
return ((n,)+more for more in pickk(k-1,N,m+1)
for n in range(m,more))

def partitionN(k,N) :
subsets = [frozenset(S) for S in pickk(k,N)]
def exhaust(rest,bound=0) :
if len(rest) < k :
if rest :
yield [sorted(rest)]
else :
yield []
for newbound in range(bound,len(subsets)) :
S = subsets[newbound]
if rest >= S :
sS = [sorted(S)]
for restpart in exhaust(rest-S,newbound+1) :
yield sS+restpart
return exhaust(set(range(N)))

partition = lambda k,S : [[[S[j] for j in S0] for S0 in P0]
for P0 in partitionN(k,len(S))]

>>> partition(2,[1,2,3,4])
[[[1, 2], [3, 4]], [[1, 3], [2, 4]], [[2, 3], [1, 4]]]
>>> partition(3,[1,2,3,4])
[[[1, 2, 3], ], [[1, 2, 4], ], [[1, 3, 4], ], [[2, 3, 4], ]]

CAVEAT : insufficiently tested, not proved correct, uncommented, provided as is

```