# Eulerian Numbers Calculator: A(n, k)

## Eulerian Numbers

A simple and efficient tool to compute Eulerian numbers for given 'n' and 'k' values.

### What are Eulerian Numbers?

Eulerian numbers are a sequence of integers that appear in combinatorics, named after the Swiss mathematician Leonhard Euler. They represent the number of permutations of the numbers from 1 to n in which exactly k elements are greater than the element immediately preceding them. The Eulerian number A(n, k) can be calculated using a recursive formula:

**A(n, k) = (n - k) * A(n - 1, k - 1) + (k + 1) * A(n - 1, k)**

with the base cases:

- A(n, 0) = 1 for n ≥ 0
- A(0, k) = 0 for k > 0

Eulerian numbers have various applications in combinatorics, including counting permutations, analyzing sorting algorithms, and studying polytopes.

### How to Use the Eulerian Numbers Calculator

Our Eulerian Numbers Calculator is a user-friendly tool that computes the Eulerian number A(n, k) for given 'n' and 'k' values. To use the calculator, follow these simple steps:

- Enter the value for 'n' in the first input field. 'n' must be a non-negative integer.
- Enter the value for 'k' in the second input field. 'k' must be a non-negative integer.
- The result, A(n, k), will be displayed below the input fields.

This calculator uses a dynamic programming approach to efficiently compute the Eulerian number A(n, k) in a short amount of time.

#### Example: Computing an Eulerian Number

Let's say we want to compute the Eulerian number A(4, 2). Here's how to use the Eulerian Numbers Calculator:

- Enter '4' in the 'n' input field.
- Enter '2' in the 'k' input field.
- The result, A(4, 2) = 5, will be displayed below the input fields.

This means that there are 5 permutations of the numbers 1 to 4 in which exactly 2 elements are greater than the element immediately preceding them.

#### Learn More About Eulerian Numbers

Eulerian numbers are a fascinating area of combinatorial mathematics. If you are interested in learning more about Eulerian numbers, their properties, and applications, consider exploring the following resources:

- Eulerian number - Wikipedia
- Eulerian Numbers: Combinatorial Applications and Identities
- A Path to Combinatorics for Undergraduates: Counting Strategies (book)

By learning more about Eulerian numbers, you can deepen your understanding of combinatorial mathematics and discover new ways to apply these concepts to real-world problems.