Partitioning Sets in Python

The task at hand is to partition a given set of elements into all possible subsets. For instance, partitioning the set [1, 2, 3] yields the following subsets:

[[1], [2], [3]]
[[1,2], [3]]
[[1], [2,3]]
[[1,3], [2]]
[[1,2,3]]

Recursive Solution

One approach to this problem is recursion. Given a partition of n-1 elements, we can extend it to create a partition of n elements by either including the nth element in one of the existing subsets or creating a new singleton subset containing only the nth element.

This recursive algorithm effectively partitions the input set while avoiding duplicate outputs and unnecessary dependencies on external libraries:

def partition(collection):
    if len(collection) == 1:
        yield [ collection ]
        return

    first = collection[0]
    for smaller in partition(collection[1:]):
        # insert `first` in each of the subpartition's subsets
        for n, subset in enumerate(smaller):
            yield smaller[:n]   [[ first ]   subset]    smaller[n 1:]
        # put `first` in its own subset 
        yield [ [ first ] ]   smaller

something = list(range(1,5))

for n, p in enumerate(partition(something), 1):
    print(n, sorted(p))