Should help with a certain type of mathematical puzzles

digitproblems.py

import functools
import itertools
import operator

def groupings(sum_):
	if sum_ == 0:
		return [()]
	else:
		return [(i,) + subg for i in range(1, sum_ + 1) for subg in groupings(sum_ - i)]

groupings_with_length = lambda s, l: list(filter(lambda g: len(g) == l, groupings(s)))

def assign_elements(grouping, elements, fold_op=operator.add):
	element_gen = iter(elements)
	return tuple(functools.reduce(fold_op, itertools.islice(element_gen, g)) for g in grouping)

assign_digits = functools.partial(assign_elements, fold_op=lambda x, y: x*10 + y)

assert list(groupings(4)) == [(1, 1, 1, 1), (1, 1, 2), (1, 2, 1), (1, 3), (2, 1, 1), (2, 2), (3, 1), (4,)]
assert list(groupings_with_length(6, 3)) == [(1, 1, 4), (1, 2, 3), (1, 3, 2), (1, 4, 1), (2, 1, 3), (2, 2, 2), (2, 3, 1), (3, 1, 2), (3, 2, 1), (4, 1, 1)]
assert assign_elements((2, 1, 3, 4), "abcdefghij") == ("ab", "c", "def", "ghij")
assert assign_digits((1, 3, 1, 4), [1,2,3,4,5,6,7,8,9]) == (1, 234, 5, 6789)