<p>You are given an array <code>nums</code> of positive integers and a <strong>positive</strong> integer <code>k</code>.</p>

<p>A subset of <code>nums</code> is <strong>beautiful</strong> if it does not contain two integers with an absolute difference equal to <code>k</code>.</p>

<p>Return <em>the number of <strong>non-empty beautiful </strong>subsets of the array</em> <code>nums</code>.</p>

<p>A <strong>subset</strong> of <code>nums</code> is an array that can be obtained by deleting some (possibly none) elements from <code>nums</code>. Two subsets are different if and only if the chosen indices to delete are different.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> nums = [2,4,6], k = 2
<strong>Output:</strong> 4
<strong>Explanation:</strong> The beautiful subsets of the array nums are: [2], [4], [6], [2, 6].
It can be proved that there are only 4 beautiful subsets in the array [2,4,6].
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> nums = [1], k = 1
<strong>Output:</strong> 1
<strong>Explanation:</strong> The beautiful subset of the array nums is [1].
It can be proved that there is only 1 beautiful subset in the array [1].
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= nums.length &lt;= 20</code></li>
	<li><code>1 &lt;= nums[i], k &lt;= 1000</code></li>
</ul>

class Solution:
    def beautifulSubsets(self, nums: List[int], k: int) -> int:
        ans = 0

        def backtrack(cur, i): # set, int
            nonlocal ans         

            if i == len(nums):
                ans += 1
                return

            # include if remains good
            if nums[i] - k not in cur and nums[i] + k not in cur:
                cur[nums[i]] += 1
                backtrack(cur, i + 1)
                cur[nums[i]] -= 1
                if cur[nums[i]] == 0:
                    del cur[nums[i]]

            # don't include
            backtrack(cur, i + 1)

        backtrack(defaultdict(int), 0)
        return ans - 1


"""
given a beautiful array, how can I know that adding a number won't make it ugly
[2, 6], k = 2
set(2, 6), want to add 4, abs(4 - num) = 2, 6 or 2 , 4 + 2 or 4 - 2 so it's num - k or num + k

        [2,4,6] i = 0 : we can include 2 (if remains beautiful) or not include 2, 
        /     \
      [2]      [].  i = 1, if not empty then add to answer += 1
      /        /. \
    [2] i = 2. 4.  []

    might have some overlapping subproblems if non distinct elements 
    

"""
16500