You are given an array nums of positive integers and a positive integer k.

A subset of nums is considered beautiful if it does not contain any pair of elements with an absolute difference equal to k.

Return the total number of non empty beautiful subsets of nums.

Note: A subset is formed by deleting zero or more elements from nums. Two subsets are distinct if they are chosen from different indices of nums.

Constraints:

• $1 \leq$ nums.length $\leq 18$

• $1 \leq$ nums[i], k $\leq 1000$

Input: nums = [5, 5, 5]-List[int], k = 2-int
Output: 7-int
Explanation: No two elements have an absolute difference of 2, so every non-empty subset is beautiful. There are \(2^3 - 1 = 7\) such subsets.

Input: nums = [2, 3, 8, 9]-List[int], k = 1-int
Output: 8-int
Explanation: The conflicting pairs are (2, 3) and (8, 9). Each pair contributes 3 valid choices: choose neither, choose the first, or choose the second. Excluding the empty subset gives \(3 imes 3 - 1 = 8\).

Input: nums = [6, 11, 17]-List[int], k = 4-int
Output: 7-int
Explanation: No pair of elements has an absolute difference of 4, so every non-empty subset is beautiful. There are \(2^3 - 1 = 7\) such subsets.

Input: nums = [4, 7, 10, 13]-List[int], k = 3-int
Output: 7-int
Explanation: Each adjacent pair differs by 3, so no adjacent elements can be selected together. The number of valid subsets of this four-element chain is 8 including the empty subset, resulting in 7 non-empty beautiful subsets.

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