Count Number of Special Subsequences

Problem Description

Given an array nums consisting of only integers 0, 1, and 2, return the number of different subsequences that are special. A sequence is special if it consists of a positive number of 0s, followed by a positive number of 1s, and then a positive number of 2s. Return the answer modulo 10^9 + 7.

Key Insights

• A special subsequence must have at least one 0, one 1, and one 2, in that order.
• The order of 0s, 1s, and 2s in the subsequence must be preserved as per their positions in the original array.
• To count subsequences efficiently, we can use combinatorial counting based on the number of occurrences of 0, 1, and 2.
• Dynamic programming can be leveraged to keep track of valid subsequences as we iterate through the array.

Space and Time Complexity

Time Complexity: O(n)
Space Complexity: O(1)

Solution

The approach involves iterating through the array while maintaining counts of the number of 0s, 1s, and 2s encountered so far. We can use these counts to compute the number of special subsequences by considering each position of 2s and how many valid pairs of 0s and 1s can precede them.

1. Initialize counters for count0, count1, and count2 to track the occurrences of 0, 1, and 2 respectively.
2. For each 2 encountered, calculate the number of valid 0,1 pairs that can be formed using count0 and count1.
3. Use modulo 10^9 + 7 to keep the results within bounds.
4. The final result will be the sum of all valid special subsequences.

Code Solutions