<p>You are given <code>nums</code>, an array of positive integers of size <code>2 * n</code>. You must perform <code>n</code> operations on this array.</p>

<p>In the <code>i<sup>th</sup></code> operation <strong>(1-indexed)</strong>, you will:</p>

<ul>
	<li>Choose two elements, <code>x</code> and <code>y</code>.</li>
	<li>Receive a score of <code>i * gcd(x, y)</code>.</li>
	<li>Remove <code>x</code> and <code>y</code> from <code>nums</code>.</li>
</ul>

<p>Return <em>the maximum score you can receive after performing </em><code>n</code><em> operations.</em></p>

<p>The function <code>gcd(x, y)</code> is the greatest common divisor of <code>x</code> and <code>y</code>.</p>

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

<pre>
<strong>Input:</strong> nums = [1,2]
<strong>Output:</strong> 1
<strong>Explanation:</strong>&nbsp;The optimal choice of operations is:
(1 * gcd(1, 2)) = 1
</pre>

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

<pre>
<strong>Input:</strong> nums = [3,4,6,8]
<strong>Output:</strong> 11
<strong>Explanation:</strong>&nbsp;The optimal choice of operations is:
(1 * gcd(3, 6)) + (2 * gcd(4, 8)) = 3 + 8 = 11
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> nums = [1,2,3,4,5,6]
<strong>Output:</strong> 14
<strong>Explanation:</strong>&nbsp;The optimal choice of operations is:
(1 * gcd(1, 5)) + (2 * gcd(2, 4)) + (3 * gcd(3, 6)) = 1 + 4 + 9 = 14
</pre>

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

<ul>
	<li><code>1 &lt;= n &lt;= 7</code></li>
	<li><code>nums.length == 2 * n</code></li>
	<li><code>1 &lt;= nums[i] &lt;= 10<sup>6</sup></code></li>
</ul>

from math import gcd, comb as choose
from itertools import combinations as combos

class Solution:
    def maxScore(self, nums: List[int]) -> int:
        n = len(nums) // 2
        num_subsets = 4**n
        dp = [[0]*choose(2 * n, 2 * n - 2 * i) for i in range(n)]
        subset_indices = []

        for i in range(n):
            subset_indices.append({})
            for j, combo in enumerate(combos(range(2 * n), 2 * n - 2 * i)):
                subset_indices[-1][combo] = j

        for i, pair in enumerate(combos(range(2 * n), 2)):
            index = subset_indices[-1][pair]
            dp[-1][index] = n * gcd(nums[pair[0]], nums[pair[1]])

        for i in reversed(range(n - 1)):
            for k, combo in enumerate(combos(range(2 * n), 2 * n - 2 * i)):
                index = subset_indices[i][combo]
                for pair in combos(combo, 2):
                    index1 = subset_indices[i + 1][tuple(j for j in combo if j not in pair)]
                    d = gcd(nums[pair[0]], nums[pair[1]])
                    dp[i][index] = max(
                        dp[i][index],
                        dp[i + 1][index1] + (i + 1) * d
                    )

        return dp[0][0]
2185