Python Coding Test Course, Finding the Order of Permutations

Problem Description

Given a number N, we need to generate all permutations of numbers from 1 to N and find the order of a specific permutation. For example, when N=3, the possible permutations are [1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1], totaling 6, and we need to find the position of each permutation starting from index 1.

To solve this problem, we will receive the following input:

• N: The number of elements to form the permutation
• P: A specific permutation (provided in list format)

Here, the output will be the order of the given permutation P.

Example Input and Output

Input

3
[2, 3, 1]
            

Output

4
            

In the above case, the given permutation [2, 3, 1] is the 4th permutation out of a total of 6 permutations.

Problem-Solving Process

1. Generating Permutations

To solve the problem, we will first use the permutations function from Python’s itertools module to generate all permutations from 1 to N. The permutations function is very useful for returning all permutations of a given iterable.

2. Sorting into a List

The generated permutations will be stored in a list to maintain a sorted form. This is to quickly find permutations indexed from 1.

3. Finding the Index of a Specific Permutation

We will search for the given specific permutation in the list to find its index. Since the index starts from 0, we will add 1 when outputting the result.

Python Code Implementation

Now, let’s implement the Python code based on the approach described above:

from itertools import permutations

def find_permutation_index(N, P):
    # Create a list of numbers from 1 to N
    numbers = list(range(1, N+1))
    
    # Generate all permutations
    all_permutations = list(permutations(numbers))

    # Find the index of the specific permutation
    index = all_permutations.index(tuple(P)) + 1  # Add 1 to adjust to 1-based indexing
    return index

# Example execution
N = 3
P = [2, 3, 1]
print(find_permutation_index(N, P))
            

The above code defines the find_permutation_index function, which finds the index of a specific permutation given N and P. We utilize the itertools module to automatically generate all permutations and easily find the position of the permutation using the index method.

Complexity Analysis

The time complexity of this algorithm is proportional to the number of permutations generated, which is N!. While this may be an inefficient approach, it is easy to understand and implement when N is small. However, an efficient approach is needed for larger values of N.

Improved Approach

For example, we could use a mathematical approach to collect counts for each number and systematically calculate how many combinations each specific number can generate. This could be a more efficient method.

Conclusion

In this tutorial, we learned how to solve the problem of finding the order of permutations. We presented a simple implementation using Python’s itertools module, so apply it to real problems for a deeper understanding.