<p>Given an integer array <code>nums</code> and an integer <code>k</code>, return <code>true</code> if it is possible to divide this array into <code>k</code> non-empty subsets whose sums are all equal.</p>

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

<pre>
<strong>Input:</strong> nums = [4,3,2,3,5,2,1], k = 4
<strong>Output:</strong> true
<strong>Explanation:</strong> It is possible to divide it into 4 subsets (5), (1, 4), (2,3), (2,3) with equal sums.
</pre>

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

<pre>
<strong>Input:</strong> nums = [1,2,3,4], k = 3
<strong>Output:</strong> false
</pre>

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

<ul>
	<li><code>1 &lt;= k &lt;= nums.length &lt;= 16</code></li>
	<li><code>1 &lt;= nums[i] &lt;= 10<sup>4</sup></code></li>
	<li>The frequency of each element is in the range <code>[1, 4]</code>.</li>
</ul>

class Solution:
    def canPartitionKSubsets(self, nums: List[int], k: int) -> bool:

        total_sum = sum(nums)
        if total_sum % k != 0:
            return False

        k_sum = total_sum // k
        if max(nums) > k_sum:
            return False

        nums.sort(reverse=True)

        memo = {}

        n = len(nums)
        used = [False] * n
        
        def dfs(k, curr_sum, start_point):
            if k == 0:
                return True

            if curr_sum == k_sum:
                return dfs(k-1, 0, 0)
            
            memo_key = (curr_sum, tuple(used))
            if memo_key in memo:
                return memo[memo_key]

            for i in range(start_point, n):
                if used[i] or curr_sum + nums[i] > k_sum:
                    continue
                used[i] = True
                if dfs(k, curr_sum + nums[i], i+1):
                    memo[memo_key] = True
                    return True
                used[i] = False

            memo[memo_key] = False
            return False

        return dfs(k, 0, 0)
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