<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>

from typing import List

class Solution:
    def beautifulSubsets(self, nums: List[int], k: int) -> int:
        # Helper function to check if a subset is beautiful
        def is_beautiful(subset: List[int]) -> bool:
            subset_set = set(subset)
            for num in subset:
                if (num - k) in subset_set or (num + k) in subset_set:
                    return False
            return True
        
        # Recursive function to generate all subsets
        def backtrack(index: int, current: List[int]):
            if index == len(nums):
                if current and is_beautiful(current):
                    # If current subset is non-empty and beautiful, count it
                    beautiful_subsets.append(list(current))
                return
            
            # Include nums[index] in the current subset
            current.append(nums[index])
            backtrack(index + 1, current)
            current.pop()
            
            # Exclude nums[index] from the current subset
            backtrack(index + 1, current)
        
        beautiful_subsets = []
        backtrack(0, [])
        
        # Return the number of beautiful subsets
        return len(beautiful_subsets)

# Example usage:
# solution = Solution()
# print(solution.beautifulSubsets([2, 4, 6], 2))  # Output: 4
# print(solution.beautifulSubsets([1], 1))       # Output: 1
183500