The Number of Beautiful Subsets

Challenge Points: 20

Level: Medium

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

We call a subset beautiful if no two numbers inside it differ by exactly k.
Your task is to determine how many non-empty beautiful subsets can be formed from nums.

A subset is formed by choosing any combination of elements while keeping their original index distinctions. Two subsets are counted as different if they correspond to different index selections.

Example 1:

Input: nums = [3, 5, 8, 10], k = 2

Output: 8

Explanation:

Valid beautiful subsets include: [3], [5], [8], [10], [3,8], [3,10], [5,8], [5,10], [8,10], [3,5,8], [3,8,10] (For example, [3,5] is invalid because |3 - 5| = 2 = k.)