Count ways to partition a number into increasing sequences of digits

Given a numeric string S, the task is to find the number of ways to partition a string into substrings consisting of digits in increasing order.

Examples:

Input: S = "1345"
Output: 5
Explanation: Possible partitions are as follows: 

1. [1345]
2. [13, 45], [1, 345]
3. [1, 3, 45]
4. [1, 3, 4, 5]

Input: S = "12"
Output: 2

Approach: This problem can be solved by observing that between each digit either it will be a part of the previous number or it will be a new number so to solve the problem recursion can be used. Follow the steps below to solve the problem:

• Initialize an integer variable, say count as 0, to store the number of ways to partition a string into increasing subsets.
• Declare a function print() with index(storing current position), string S(given string in the question), and string ans( as parameters.
• Now, following two cases are required to be considered: 
  
  • If S[index] is inserted in the previous number, then append S[index] at the end of ans and recall the function print() with parameters index + 1, S, and ans.
  • If S[index] is not a part of the previous number, then append " "(space) at the end of ans and then insert S[index] and recall the function print() with parameters index + 1, S, ans.
• If index = S.length(), then check if  the digits in the sequences formed are in increasing order or not. If the sequences formed are increasing, increase count by 1.
• Print count as the answer after performing the above steps.

Below is the implementation of the above approach:

5

Time Complexity: O(N*2^N) where N is the length of string S
Auxiliary Space: O(1)