Counting Gessel Walks

Introduction

In the realm of combinatorics, a fascinating problem has been studied extensively, known as the Gessel walk. This problem involves counting the number of walks on a square lattice that start and end at the origin, with specific step constraints. The Gessel walk has been a subject of interest in mathematics, particularly in the field of combinatorics, and has also been explored in the context of code golf. In this article, we will delve into the world of Gessel walks, exploring their definition, properties, and counting methods.

What is a Gessel Walk?

A Gessel walk is a walk on the square lattice that starts and ends at the origin. The possible steps in a Gessel walk are:

• Right step: (1,0)
• Left step: (-1,0)
• Up step: (1,1)
• Down step: (-1,-1)

These steps are the building blocks of a Gessel walk, and the sequence of steps determines the path taken by the walk.

Properties of Gessel Walks

Gessel walks have several interesting properties that make them a fascinating subject of study. Some of these properties include:

• Symmetry: Gessel walks exhibit symmetry under reflection about the x-axis and y-axis.
• Reversibility: A Gessel walk can be reversed by simply reversing the order of its steps.
• Combinatorial structure: Gessel walks can be represented as a sequence of steps, which can be analyzed using combinatorial techniques.

Counting Gessel Walks

The main problem of interest in the context of Gessel walks is counting the number of walks of a given length. This problem is denoted as $g(n)$, where $n$ is the length of the walk. The sequence $g(n)$ is known as the Gessel walk sequence, and it has been extensively studied in the field of combinatorics.

The Gessel Walk Sequence

The Gessel walk sequence is a sequence of numbers that count the number of Gessel walks of a given length. The sequence is defined as follows:

$$g(n) = \sum_{k=0}^{n} \binom{n}{k}^2$$

where $\binom{n}{k}$ is the binomial coefficient, which represents the number of ways to choose $k$ elements from a set of $n$ elements.

Computing the Gessel Walk Sequence

Computing the Gessel walk sequence is a challenging problem, and several methods have been proposed to compute the sequence. Some of these methods include:

• Recursion: The Gessel walk sequence can be computed recursively using the following formula:

$$g(n) = \sum_{k=0}^{n-1} g(k) \cdot g(n-1-k)$$

• Dynamic programming: The Gessel walk sequence can be computed using dynamic programming techniques, which involve breaking down the problem into smaller sub-problems and solving each sub-problem only once.

Code Golf and Gessel Walks

Code golf is a competitive programming community that focuses on writing the shortest possible code to solve a given problem. Gessel walks have been a subject of interest in code golf, and several code golf challenges have been proposed to compute the Gessel walk sequence.

Example Code

Here is an example of code in Python that computes the Gessel walk sequence using dynamic programming:

def gessel_walk(n):
    g = [0] * (n + 1)
    g[0] = 1
    for i in range(1, n + 1):
        for j in range(i):
            g[i] += g[j] * g[i - 1 - j]
    return g[n]
print(gessel_walk(10))

This code computes the Gessel walk sequence up to length 10 and prints the result.

Conclusion

In conclusion, Gessel walks are a fascinating subject of study in combinatorics and code golf. The Gessel walk sequence is a sequence of numbers that count the number of Gessel walks of a given length, and it has been extensively studied in the field of combinatorics. Computing the Gessel walk sequence is a challenging problem, and several methods have been proposed to compute the sequence. Code golf has also been a subject of interest in the context of Gessel walks, and several code golf challenges have been proposed to compute the Gessel walk sequence.

References

• OEIS A135404: The Gessel walk sequence is listed in the Online Encyclopedia of Integer Sequences (OEIS) under the sequence number A135404.
• Gessel, I. M.: The Gessel walk was first introduced by I. M. Gessel in his 1992 paper "A Combinatorial Formula for the Number of Lattice Paths from (0,0) to (n,n)".

Further Reading

For further reading on Gessel walks and combinatorics, we recommend the following resources:

• "Combinatorics: Topics, Techniques, Algorithms" by Peter J. Cameron: This book provides an introduction to combinatorics and covers topics such as permutations, combinations, and lattice paths.
• "The Art of Combinatorics" by Miklós Bóna: This book provides an introduction to combinatorics and covers topics such as permutations, combinations, and lattice paths.

Q: What is a Gessel walk?

A: A Gessel walk is a walk on the square lattice that starts and ends at the origin, with possible steps (1,0), (-1,0), (1,1), and (-1,-1).

Q: What is the Gessel walk sequence?

A: The Gessel walk sequence is a sequence of numbers that count the number of Gessel walks of a given length. The sequence is defined as follows:

$$g(n) = \sum_{k=0}^{n} \binom{n}{k}^2$$

Q: How do I compute the Gessel walk sequence?

A: There are several methods to compute the Gessel walk sequence, including:

• Recursion: The Gessel walk sequence can be computed recursively using the following formula:

$$g(n) = \sum_{k=0}^{n-1} g(k) \cdot g(n-1-k)$$

• Dynamic programming: The Gessel walk sequence can be computed using dynamic programming techniques, which involve breaking down the problem into smaller sub-problems and solving each sub-problem only once.

Q: What is the significance of the Gessel walk sequence?

A: The Gessel walk sequence has several interesting properties and applications, including:

• Symmetry: The Gessel walk sequence exhibits symmetry under reflection about the x-axis and y-axis.
• Reversibility: The Gessel walk sequence can be reversed by simply reversing the order of its steps.
• Combinatorial structure: The Gessel walk sequence can be represented as a sequence of steps, which can be analyzed using combinatorial techniques.

Q: How do I use the Gessel walk sequence in code golf?

A: The Gessel walk sequence can be used in code golf to solve problems related to lattice paths and combinatorics. Here is an example of code in Python that computes the Gessel walk sequence using dynamic programming:

def gessel_walk(n):
    g = [0] * (n + 1)
    g[0] = 1
    for i in range(1, n + 1):
        for j in range(i):
            g[i] += g[j] * g[i - 1 - j]
    return g[n]
print(gessel_walk(10))

Q: What are some real-world applications of the Gessel walk sequence?

A: The Gessel walk sequence has several real-world applications, including:

• Lattice path problems: The Gessel walk sequence can be used to solve lattice path problems, which involve finding the number of paths between two points in a lattice.
• Combinatorial optimization: The Gessel walk sequence can be used to solve combinatorial optimization problems, which involve finding the optimal solution to a problem that involves counting or arranging objects.

Q: How do I learn more about Gessel walks and combinatorics?

A: There are several resources available to learn more about Gessel walks and combinatorics, including:

• Books: There are several books available on combinatorics and lattice paths, including "Combinatorics: Topics, Techniques, Algorithms" by Peter J. Cameron and "The Art of Combinatorics" by Miklós Bóna.
• Online resources: There are several online resources available on combinatorics and lattice paths, including the Online Encyclopedia of Integer Sequences (OEIS) and the Combinatorics subreddit.

Q: What are some common mistakes to avoid when working with Gessel walks?

A: Some common mistakes to avoid when working with Gessel walks include:

• Incorrectly counting the number of steps: Make sure to correctly count the number of steps in a Gessel walk.
• Failing to account for symmetry: Make sure to account for symmetry when computing the Gessel walk sequence.
• Using incorrect formulas: Make sure to use the correct formulas when computing the Gessel walk sequence.

Q: How do I get started with Gessel walks and combinatorics?

A: To get started with Gessel walks and combinatorics, follow these steps:

1. Learn the basics of combinatorics: Start by learning the basics of combinatorics, including permutations, combinations, and lattice paths.
2. Understand the Gessel walk sequence: Learn about the Gessel walk sequence and how it is computed.
3. Practice computing the Gessel walk sequence: Practice computing the Gessel walk sequence using dynamic programming and recursion.
4. Explore real-world applications: Explore real-world applications of the Gessel walk sequence, including lattice path problems and combinatorial optimization.