# Permutation Unrank Calculator

## Permutation Unrank

Welcome to the Permutation Unrank Calculator, a useful tool that helps you find the permutation corresponding to a given rank in lexicographic order and a specified number of elements. Whether you're a student, a researcher, or a mathematician, this calculator will make it easy for you to explore the fascinating world of combinatorics and permutations.

### What is Permutation Unrank?

Permutation unrank is the process of finding a unique permutation for a given rank (position) in the lexicographic order and a specified number of elements. In other words, it is a method of converting a rank into a corresponding permutation, considering all possible permutations of the given elements sorted in lexicographic order. This process is often used in combinatorial algorithms, cryptography, and other fields that require working with permutations.

For example, suppose you have three elements, 0, 1, and 2, and you want to find the permutation corresponding to the rank 4. The lexicographically ordered permutations of these elements are:

- [0, 1, 2]
- [0, 2, 1]
- [1, 0, 2]
- [1, 2, 0]
- [2, 0, 1]
- [2, 1, 0]

The permutation unrank process allows you to determine that the permutation corresponding to rank 4 is [2, 0, 1].

### How it works:

The Permutation Unrank Calculator uses an algorithm based on the factoradic (factorial number system) to generate the desired permutation. This algorithm is more procedural than a simple mathematical formula but effectively solves the Permutation Unrank problem.

Here's a high-level description of the algorithm:

- Initialize an empty result list and a list containing elements from 0 to n - 1, where n is the number of elements.
- For i from 1 to n:
- Compute the factorial of n - i.
- Divide the current rank by the factorial, and obtain the quotient and remainder.
- Use the quotient as an index to get the element from the list of remaining elements.
- Append the selected element to the result list.
- Remove the selected element from the list of remaining elements.
- Update the current rank to the remainder obtained in step b.

- The result list is the permutation corresponding to the given rank.

### Instructions:

To use the Permutation Unrank Calculator, follow these simple steps:

- Enter the number of elements (n) in the input field.
- Enter the rank (r) of the desired permutation in the input field.
- The calculator will display the permutation corresponding to the given rank and number of elements.

#### Example:

Let's say you want to find the permutation corresponding to rank 4 with 3 elements.

- Enter the number of elements (n): 3
- Enter the rank (r): 4
- The calculator will display the permutation: 1, 0, 2

The Permutation Unrank Calculator is a powerful and easy-to-use tool that helps you explore the world of permutations and combinatorics. By providing the ability to generate permutations based on a given rank and a specified number of elements, this calculator is an invaluable resource for students, researchers, and mathematicians alike.