<p>You are given an array <code>nums</code> consisting of <strong>positive</strong> integers and an integer <code>k</code>.</p>

<p><strong>Partition</strong> the array into two ordered <strong>groups</strong> such that each element is in exactly <strong>one</strong> group. A partition is called great if the <strong>sum</strong> of elements of each group is greater than or equal to <code>k</code>.</p>

<p>Return <em>the number of <strong>distinct</strong> great partitions</em>. Since the answer may be too large, return it <strong>modulo</strong> <code>10<sup>9</sup> + 7</code>.</p>

<p>Two partitions are considered distinct if some element <code>nums[i]</code> is in different groups in the two partitions.</p>

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

<pre>
<strong>Input:</strong> nums = [1,2,3,4], k = 4
<strong>Output:</strong> 6
<strong>Explanation:</strong> The great partitions are: ([1,2,3], [4]), ([1,3], [2,4]), ([1,4], [2,3]), ([2,3], [1,4]), ([2,4], [1,3]) and ([4], [1,2,3]).
</pre>

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

<pre>
<strong>Input:</strong> nums = [3,3,3], k = 4
<strong>Output:</strong> 0
<strong>Explanation:</strong> There are no great partitions for this array.
</pre>

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

<pre>
<strong>Input:</strong> nums = [6,6], k = 2
<strong>Output:</strong> 2
<strong>Explanation:</strong> We can either put nums[0] in the first partition or in the second partition.
The great partitions will be ([6], [6]) and ([6], [6]).
</pre>

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

<ul>
	<li><code>1 &lt;= nums.length, k &lt;= 1000</code></li>
	<li><code>1 &lt;= nums[i] &lt;= 10<sup>9</sup></code></li>
</ul>

MOD = 1e9 + 7
class Solution:
    def countPartitions(self, nums: List[int], k: int) -> int:
        MOD = 10**9 + 7
        n = len(nums)
        total_sum = sum(nums)
        
        # If the total sum of nums is less than 2 * k, no valid partition exists
        if total_sum < 2 * k:
            return 0
        
        # Memoization dictionary
        memo = {}
        
        # Recursive function to count subsets with sum less than k
        def countSubsets(i, curr_sum):
            if curr_sum >= k:
                return 0
            if i == n:
                return 1  # A valid subset with sum less than k
            
            if (i, curr_sum) in memo:
                return memo[(i, curr_sum)]
            
            # Two choices: include nums[i] in the current subset or exclude it
            include = countSubsets(i + 1, curr_sum + nums[i])
            exclude = countSubsets(i + 1, curr_sum)
            
            # Store result in memo
            memo[(i, curr_sum)] = (include + exclude) % MOD
            return memo[(i, curr_sum)]
        
        # Total number of subsets is 2^n
        total_subsets = pow(2, n, MOD)
        
        # Subtract the number of subsets where at least one partition sum is less than k
        valid_subsets = total_subsets - 2 * countSubsets(0, 0)
        
        return valid_subsets % MOD

1509