Problem Statement
Count Vowels Permutation
Given an integer n, your task is to count how many strings of length n can be formed under the following rules:
- Each character is a lower case vowel (
'a','e','i','o','u') - Each vowel
'a'may only be followed by an'e'. - Each vowel
'e'may only be followed by an'a'or an'i'. - Each vowel
'i'may not be followed by another'i'. - Each vowel
'o'may only be followed by an'i'or a'u'. - Each vowel
'u'may only be followed by an'a'.
Since the answer may be too large, return it modulo 10^9 + 7.
Examples
Example 1:
Input: n = 1
Output: 5
Explanation: All possible strings are: 'a', 'e', 'i', 'o', 'u'.
Example 2:
Input: n = 2
Output: 10
Explanation: All possible strings are: 'ae', 'ea', 'ei', 'ia', 'ie', 'io', 'iu', 'oi', 'ou', 'ua'.
Example 3:
Input: n = 5
Output: 68
Explanation: This is the number of possible strings of length 5 that can be formed under the given rules.
Constraints
- 1 <= n <= 2 * 10^4
Problem Breakdown
To solve this problem, we need to:
- This problem can be solved using dynamic programming.
- We need to keep track of the count of strings ending with each vowel.
- The state of the DP array is defined by the current length and the last vowel.
- The recurrence relation involves considering all possible previous vowels that can lead to the current vowel.
- The final answer is the sum of the counts for all vowels at length n.
String Problems
Learning Objective
Apply string manipulation concepts to solve a real-world problem.
Count Vowels Permutation
Given an integer n, your task is to count how many strings of length n can be formed under the following rules:
- Each character is a lower case vowel (
'a','e','i','o','u') - Each vowel
'a'may only be followed by an'e'. - Each vowel
'e'may only be followed by an'a'or an'i'. - Each vowel
'i'may not be followed by another'i'. - Each vowel
'o'may only be followed by an'i'or a'u'. - Each vowel
'u'may only be followed by an'a'.
Since the answer may be too large, return it modulo 10^9 + 7.
Examples
Example 1:
All possible strings are: 'a', 'e', 'i', 'o', 'u'.
Example 2:
All possible strings are: 'ae', 'ea', 'ei', 'ia', 'ie', 'io', 'iu', 'oi', 'ou', 'ua'.
Example 3:
This is the number of possible strings of length 5 that can be formed under the given rules.
Problem Breakdown
Constraints:
- 1 <= n <= 2 * 10^4
Key Insights:
This problem can be solved using dynamic programming.
We need to keep track of the count of strings ending with each vowel.
The state of the DP array is defined by the current length and the last vowel.
The recurrence relation involves considering all possible previous vowels that can lead to the current vowel.
The final answer is the sum of the counts for all vowels at length n.
Real-World Application
This problem has several practical applications:
Language Processing
Analyzing and generating text with specific phonetic or grammatical constraints.
Combinatorial Problems
Solving problems that involve counting the number of ways to arrange objects under certain constraints.
Markov Chains
Modeling systems where the next state depends only on the current state, not on the sequence of events that preceded it.