"""
Given an integer array with all positive numbers and no duplicates, find the number of possible combinations that add up
to a positive integer target.

Example:

nums = [1, 2, 3]
target = 4

The possible combination ways are:
(1, 1, 1, 1)
(1, 1, 2)
(1, 2, 1)
(1, 3)
(2, 1, 1)
(2, 2)
(3, 1)

Note that different sequences are counted as different combinations.

Therefore the output is 7.
Follow up:
What if negative numbers are allowed in the given array?
How does it change the problem?
What limitation we need to add to the question to allow negative numbers?
"""
__author__ = 'Daniel'


class Solution(object):
    def combinationSum4(self, nums, target):
        """
        Let F[i] be the number of combinations ways for number i
        F[i] = sum(F[i-k] for k in nums)

        If negative number allowed, [1, -1], 1, then infinite combinations.
        If the target is large, recursion & memoization is more space saving
        :type nums: List[int]
        :type target: int
        :rtype: int
        """
        F = [0 for _ in xrange(target + 1)]
        nums = filter(lambda x: x <= target, nums)
        for k in nums:
            F[k] = 1

        for i in xrange(target + 1):
            for k in nums:
                if i - k >= 0:
                    F[i] += F[i-k]

        return F[target]


if __name__ == "__main__":
    assert Solution().combinationSum4([1, 2, 3], 4) == 7

"""
Given an integer array with all positive numbers and no duplicates, find the number of possible combinations that add up
to a positive integer target.

Example:

nums = [1, 2, 3]
target = 4

The possible combination ways are:
(1, 1, 1, 1)
(1, 1, 2)
(1, 2, 1)
(1, 3)
(2, 1, 1)
(2, 2)
(3, 1)

Note that different sequences are counted as different combinations.

Therefore the output is 7.
Follow up:
What if negative numbers are allowed in the given array?
How does it change the problem?
What limitation we need to add to the question to allow negative numbers?
"""
__author__ = 'Daniel'


class Solution(object):
    def combinationSum4(self, nums, target):
        """
        Let F[i] be the number of combinations ways for number i
        F[i] = sum(F[i-k] for k in nums)

        If negative number allowed, [1, -1], 1, then infinite combinations.
        If the target is large, recursion & memoization is more space saving
        :type nums: List[int]
        :type target: int
        :rtype: int
        """
        F = [0 for _ in xrange(target + 1)]
        nums = filter(lambda x: x <= target, nums)
        for k in nums:
            F[k] = 1

        for i in xrange(target + 1):
            for k in nums:
                if i - k >= 0:
                    F[i] += F[i-k]

        return F[target]


if __name__ == "__main__":
    assert Solution().combinationSum4([1, 2, 3], 4) == 7

Given an integer array with all positive numbers and no duplicates, find the number of possible combinations that add up to a positive integer target.  Example:  nums = [1, 2, 3] target = 4  The possible combination ways are: (1, 1, 1, 1) (1, 1, 2) (1, 2, 1) (1, 3) (2, 1, 1) (2, 2) (3, 1)  Note that different sequences are counted as different combinations.  Therefore the output is 7. Follow up: What if negative numbers are allowed in the given array? How does it change the problem? What limitation we need to add to the question to allow negative numbers? Let F[i] be the number of combinations ways for number i         F[i] = sum(F[i-k] for k in nums)          If negative number allowed, [1, -1], 1, then infinite combinations.         If the target is large, recursion & memoization is more space saving         :type nums: List[int]         :type target: int         :rtype: int