<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 __init__(self):
        self.mem = {}


    def canPartitionKSubsets(self, nums: List[int], k: int) -> bool:
        nums_sum = sum(nums)
        visited = set()
        return self.can_partition_k_subsets(sorted(nums), k, sum(nums) / k, 0, visited, 0)        


    def can_partition_k_subsets(self, nums: List[int], k: int, target: int, cur_sum: int, visited: set[int], start_from: int) -> bool:
        # print(f'{nums} {k} {target} {cur_sum} {visited}')
        key = self._serialize(nums, visited)
        if key in self.mem:
            return self.mem[key]
        if k == 0 and len(visited) == len(nums):
            return True
        i = start_from + 1
        if i >= len(nums):
                i = 0    
        while i != start_from:
            
            if i not in visited:
                if cur_sum + nums[i] == target:
                    visited.add(i)
                    if self.can_partition_k_subsets(nums, k-1, target, 0, visited, i):
                        return True
                    else:
                        visited.remove(i)
                if cur_sum + nums[i] < target:
                    visited.add(i)
                    if self.can_partition_k_subsets(nums, k, target, cur_sum + nums[i], visited, i):
                        return True
                    else:
                        visited.remove(i)
            i += 1
            if i >= len(nums):
                i = 0           
        self.mem[key] = False    
        return False

    def _serialize(self, nums: List[int], visited: set[int]) -> str:
        return ''.join(['1' if i in visited else '0' for i in range(len(nums))])
        
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