<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 {
public:
    int dfs(vector<int> &nums, int idx, int mask, int k, vector<vector<int>> &dp) {
        int n = nums.size();
        if (idx == n) return 1; // Base case: consider all elements

        if (dp[idx][mask] != -1) return dp[idx][mask]; // Return memoized result

        // Option 1: Do not take the current element
        int notTaken = dfs(nums, idx + 1, mask, k, dp);

        // Option 2: Take the current element
        bool canTake = true;
        for (int j = 0; j < n; ++j) {
            if ((mask & (1 << j)) && (abs(nums[idx] - nums[j]) == k)) {
                canTake = false;
                break;
            }
        }

        int taken = 0;
        if (canTake) {
            taken = dfs(nums, idx + 1, mask | (1 << idx), k, dp);
        }

        return dp[idx][mask] = taken + notTaken; // Store and return result
    }

    int beautifulSubsets(vector<int> &nums, int k) {
        int n = nums.size();
        vector<vector<int>> dp(n + 1, vector<int>(1 << n, -1)); // DP table
        int result = dfs(nums, 0, 0, k, dp);
        return result - 1; // Subtract 1 to exclude the empty subset
    }
};

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