377. Combination Sum IV——DP本质：针对结果的迭代，dp[ans] <= dp[ans-i] & dp[i] 找三者关系 思考问题的维度+1，除了数据集迭代还有考虑结果

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?

class Solution(object):
    def combinationSum4(self, nums, target):
        """
        :type nums: List[int]
        :type target: int
        :rtype: int
nums = [1, 2, 3]
target = 4

f(4) = [1,f(3)] if 1 in nums U [2, f(2)] if 2 in nums U [3, f(1)] if 3 in nums U [4, f(0)] if 4 in nums

The possible combination ways are:
(1, 1, 1, 1)
(1, 1, 2)
(1, 2, 1)
(1, 3)
(2, 1, 1)
(2, 2)
(3, 1)        
        """
        dp = [1] + [0]*target
        for x in range(1, target+1):
            for n in nums:
                if x >= n:
                    dp[x] += dp[x-n]
        return dp[target]