Find The Number of Good Permutations in Java
Good permutations are an intriguing subset of combinatorial mathematics. They are often used in mathematical puzzles, algorithms, and optimization problems. They satisfy predefined criteria and properties and require analytical techniques, algorithmic design, and creative problem-solving skills.
Example 1:
Input : N = 4, K = 1
Output: 1
Explanation: {1, 2, 3, 4} is the only possible good permutation.
Using Mathematical Approach
The code demonstrates a method for calculating the count of good permutations based on two parameters: n and k. It initializes the variable 'goodPermutations' to 1, adds fixed points based on 'excludedIndices,' and uses combinatorial principles to ensure that at least 'totalElements—excludedIndices' indices are moved in each permutation.
Here's an implementation of finding the number of good permutations using a mathematical approach :
FileName: GoodPermutationsCalculator.java
public class GoodPermutationsCalculator { // Method to calculate the count of good permutations public static int countGoodPermutations(int totalElements, int excludedIndices) { int goodPermutations = 1; // If there are at least 2 excluded indices, add permutations with 2 fixed points if (excludedIndices >= 2) goodPermutations += (totalElements * (totalElements - 1)) / 2; // If there are at least 3 excluded indices, add permutations with 3 fixed points if (excludedIndices >= 3) goodPermutations += (totalElements * (totalElements - 1) * (totalElements - 2)) / 2; // If there are at least 4 excluded indices, add permutations with 4 fixed points if (excludedIndices >= 4) goodPermutations += (totalElements * (totalElements - 1) * (totalElements - 2) * (totalElements - 3)) / 8 * 9; return goodPermutations; } public static void main(String[] args) { int n = 4, k = 1; System.out.println(countGoodPermutations(n, k)); } }
Output:
1
Complexity analysis: The code is simple and efficient, calculating the count of good permutations with fixed points based on `excludedIndices.` It does not use loops or iterative operations, and its time complexity is constant, denoted as O(1). It also has a constant space complexity, O(1), as it uses only a constant amount of memory without additional data structures or variables.