<p>You are given an integer array <code>nums</code>​​​ and an integer <code>k</code>. You are asked to distribute this array into <code>k</code> subsets of <strong>equal size</strong> such that there are no two equal elements in the same subset.</p>

<p>A subset&#39;s <strong>incompatibility</strong> is the difference between the maximum and minimum elements in that array.</p>

<p>Return <em>the <strong>minimum possible sum of incompatibilities</strong> of the </em><code>k</code> <em>subsets after distributing the array optimally, or return </em><code>-1</code><em> if it is not possible.</em></p>

<p>A subset is a group integers that appear in the array with no particular order.</p>

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

<pre>
<strong>Input:</strong> nums = [1,2,1,4], k = 2
<strong>Output:</strong> 4
<strong>Explanation:</strong> The optimal distribution of subsets is [1,2] and [1,4].
The incompatibility is (2-1) + (4-1) = 4.
Note that [1,1] and [2,4] would result in a smaller sum, but the first subset contains 2 equal elements.</pre>

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

<pre>
<strong>Input:</strong> nums = [6,3,8,1,3,1,2,2], k = 4
<strong>Output:</strong> 6
<strong>Explanation:</strong> The optimal distribution of subsets is [1,2], [2,3], [6,8], and [1,3].
The incompatibility is (2-1) + (3-2) + (8-6) + (3-1) = 6.
</pre>

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

<pre>
<strong>Input:</strong> nums = [5,3,3,6,3,3], k = 3
<strong>Output:</strong> -1
<strong>Explanation:</strong> It is impossible to distribute nums into 3 subsets where no two elements are equal in the same subset.
</pre>

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

<ul>
	<li><code>1 &lt;= k &lt;= nums.length &lt;= 16</code></li>
	<li><code>nums.length</code> is divisible by <code>k</code></li>
	<li><code>1 &lt;= nums[i] &lt;= nums.length</code></li>
</ul>

class Solution:
    def minimumIncompatibility(self, nums: List[int], k: int) -> int:
        '''
        if len(nums)%k!=0:
            return -1
        d = len(nums)//k
        if d == 1:
            return 0
        @cache
        def dp(xs):
            if not xs:
                return 0
            res = float('inf')
            for combo in combinations(set(xs), d):
                left, minimum, maximum = list(xs), float('inf'), float('-inf')
                for x in combo:
                    left.remove(x)
                    minimum, maximum = min(x, minimum), max(x, maximum)
                res = min(res, maximum - minimum + dp(tuple(left)))
            return res
        res = dp(tuple(nums))
        return res if res!=float('inf') else -1
        '''
        n = len(nums)
        if n%k!=0:
            return -1
        d = n//k
        if d == 1:
            return 0
        @cache
        def dfs(state):
            if state == (1<<n) - 1:
                return 0
            memo = {}
            for i in range(n):
                if ((1<<i)&state) == 0:
                    memo[nums[i]] = i
            res = float('inf')
            for it in itertools.combinations(memo.keys(), d):
                nstate = 0
                min_val, max_val = float('inf'), float('-inf')
                for x in it:
                    nstate |= (1<<memo[x])
                    min_val, max_val = min(x, min_val), max(x, max_val)
                res = min(res, max_val-min_val+dfs(state+nstate))
            return res
        res = dfs(0)
        return -1 if res == float('inf') else res

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