Number of Great Partitions

Problem Description

You are given an array nums consisting of positive integers and an integer k. Partition the array into two ordered groups such that each element is in exactly one group. A partition is called great if the sum of elements of each group is greater than or equal to k. Return the number of distinct great partitions. Since the answer may be too large, return it modulo 10^9 + 7.

Key Insights

• Each element of the array must be placed in one of the two groups.
• The sum of elements in both groups must be at least k.
• The problem can be approached using combinatorial counting or dynamic programming.
• The challenge lies in efficiently calculating the number of valid partitions while avoiding duplicates.

Space and Time Complexity

Time Complexity: O(n * k)
Space Complexity: O(k)

Solution

To solve the problem, we can use dynamic programming to count the number of ways to achieve valid partitions. We can maintain a DP array where dp[j] represents the number of ways to achieve a sum j using the elements of the array.

1. Initialize a DP array of size k + 1 with dp[0] = 1, as there is one way to achieve a sum of 0 (using no elements).
2. For each number in the nums array, iterate through the DP array backwards and update the counts based on the current number.
3. After processing all numbers, count the valid partitions by checking combinations of sums that meet the criteria of being greater than or equal to k.
4. Return the count of distinct partitions modulo 10^9 + 7.

Code Solutions