91Ó°ÊÓ

Your formula for the Catalan Number can be expressed as a binomial coefficient divided by an integer. Whenever we have a formula that calls for division by an integer, an ideal combinatorial explanation of the formula is one that uses the quotient principle. The purpose of this problem is to find such an explanation using diagonal lattice paths. \({ }^{a}\) A diagonal lattice path that never goes below the \(y\) -coordinate of its first point is called a Dyck Path. We will call a Dyck Path from (0,0) to \((2 n, 0)\) a Catalan Path of length \(2 n\). Thus the number of Catalan Paths of length \(2 n\) is the Catalan Number \(C_{n}\) (a) If a Dyck Path has \(n\) steps (each an upstep or downstep), why do the first \(k\) steps form a Dyck Path for each nonnegative \(k \leq n ?\) (b) Thought of as a curve in the plane, a diagonal lattice path can have many local maxima and minima, and can have several absolute maxima and minima, that is, several highest points and several lowest points. What is the \(y\) -coordinate of an absolute minimum point of a Dyck Path starting at (0,0)\(?\) Explain why a Dyck Path whose rightmost absolute minimum point is its last point is a Catalan Path. (h) (c) Let \(D\) be the set of all diagonal lattice paths from (0,0) to \((2 n, 0)\). (Thus these paths can go below the \(x\) -axis.) Suppose we partition \(D\) by letting \(B_{i}\) be the set of lattice paths in \(D\) that have \(i\) upsteps (perhaps mixed with some downsteps) following the last absolute minimum. How many blocks does this partition have? Give a succinct description of the block \(B_{0} \cdot(\mathrm{h})\) (d) How many upsteps are in a Catalan Path? (e) We are going to give a bijection between the set of Catalan Paths and the block \(B_{i}\) for each \(i\) between 1 and \(n\). For now, suppose the value of \(i,\) while unknown, is fixed. We take a Catalan path and break it into three pieces. The piece \(F\) (for "front") consists of all steps before the \(i\) th upstep in the Catalan path. The piece \(U\) (for "up") consists of the \(i\) th upstep. The piece \(B\) (for "back") is the portion of the path that follows the \(i\) th upstep. Thus we can think of the path as \(F U B\). Show that the function that takes \(F U B\) to \(B U F\) is a bijection from the set of Catalan Paths onto the block \(B_{i}\) of the partition. (Notice that \(B\) UF can go below the \(x\) axis.) \((\mathrm{h})\) (f) Explain how you have just given another proof of the formula for the Catalan Numbers.

Step by step solution

01

- Understanding Dyck Paths

A Dyck Path is a path on a grid that starts and ends on the x-axis and does not go below it. For a Dyck Path with n steps (each step being either an upstep or downstep), the path formed by the first k steps (where k ≤ n) remains a Dyck Path because no intermediate point can go below the x-axis. Each segment preserves this property by starting and returning to or above the previous highest point.

02

- Absolute Minimum Point of Dyck Paths

In a Dyck Path that starts at (0,0), the y-coordinate of an absolute minimum point is 0 since it cannot go below the x-axis. If a Dyck Path's last point at coordinates (2n, 0) is the rightmost absolute minimum, it means the path exactly fits the definition of a Catalan Path, which is characterized by having all steps contained within the first and last points, mimicking the property that it never goes below the x-axis.

03

- Partitioning all Diagonal Lattice Paths

Let D be the set of all diagonal lattice paths from (0,0) to (2n,0), including paths that can dip below the x-axis. By partitioning these paths based on the number of upsteps after the last absolute minimum, each subset B_i (for i=0 to n) represents paths with exactly i upsteps after the last absolute minimum. Therefore, there are n + 1 blocks in this partitioning.

04

- Description of Block B_0

The block B_0 consists of paths that have their last absolute minimum right at the end of the path, at (2n,0). This means all points beyond this minimum must be downsteps. Hence, B_0 paths will have all upsteps before a series of downsteps leading to the endpoint.

05

- Counting Upsteps in a Catalan Path

A Catalan Path has exactly n upsteps because it consists of 2n steps overall, half of which are upsteps and the other half are downsteps to ensure that the path returns to the x-axis.

06

- Bijection Between Catalan Paths and Blocks B_i

Consider a Catalan Path segmented into F (front steps before the i-th upstep), U (the i-th upstep), and B (steps after the i-th upstep). By rearranging these segments as B U F, we create a path belonging to the block B_i. This transformation is a bijection because every unique Catalan Path uniquely maps to a path within the defined block, and anytime a block path is taken, it can be reordered to form a valid Catalan Path, confirming the bijection.

07

- Proof of Catalan Numbers Formula

The bijection shown establishes a one-to-one correspondence between Catalan Paths and partition blocks B_i, for i between 1 to n. This confirms that the count of Catalan Paths corresponds directly to the combinatorial arrangement given by the Catalan Numbers, reinforcing the formula C_n for Catalan Numbers.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dyck Paths

Understanding Dyck Paths is essential for grasping the concept of Catalan Numbers. A Dyck Path is a sequence of steps on a grid that starts and ends on the x-axis (usually at (0, 0) and (2n, 0)) and never falls below the x-axis. Each step in this path is either an 'upstep' (an upward move) or a 'downstep' (a downward move). The structure of Dyck Paths ensures that at no point does the path go below its starting point on the x-axis.
A key property of Dyck Paths is that any initial segment of the path (consisting of the first k steps where k ≤ n) is also a Dyck Path. This is because the path's construction inherently avoids dropping below the x-axis, no matter how many or few steps are taken.
These specific paths can represent various combinatorial structures, making them highly valuable in solving complex enumeration problems.

Lattice Paths

Lattice paths are sequences of steps that follow a structured path on a grid of points. These steps can move in different directions, like up, down, left, or right. For problems involving Catalan Numbers, diagonal lattice paths which often use upsteps and downsteps are crucial.
In the context of Catalan Numbers, we focus on lattice paths from (0,0) to (2n,0). These paths can travel above or even below the x-axis. However, for a path to be a Dyck Path or a Catalan Path, it must maintain specific properties, such as staying above the x-axis.
Understanding lattice paths helps you appreciate the versatility of these mathematical tools. They’re used to model numerous real-life and theoretical scenarios.

Combinatorial Proofs

Combinatorial proofs are a method of demonstrating the validity of a mathematical statement by using a counting argument. Rather than resorting to algebraic manipulations or calculus, combinatorial proofs show that two sets have the same number of elements by establishing a one-to-one correspondence between them.
In discussions about Catalan Numbers, combinatorial proofs can illustrate why a given formula holds true. For instance, by showing that the set of all valid paths (like Dyck Paths) can be counted in multiple, equivalent ways, one can derive important relationships and formulas.
Combinatorial proofs are impactful because they often provide deeper insight into why certain relationships exist, rather than just confirming that they do.

Bijective Functions

A bijective function, or bijection, is a pairing between elements of two sets where each element in one set corresponds to exactly one element in the other, and vice versa. In other words, every element has a unique match, and no element is left unpaired.
For example, creating a bijection between Catalan Paths and certain blocks of other paths can demonstrate the equality of their cardinalities. When solving problems involving these paths, rearranging pieces of paths (like in the bijection described with segments F, U, and B) showcases this unique correspondence.
The power of bijective functions lies in their ability to simplify complex counting problems, transforming them into more manageable parts.

Quotient Principle

The Quotient Principle is another combinatorial tool used to count objects by dividing them into distinguishable groups. When an enumeration involves a formula with a division, the Quotient Principle provides a natural explanation.
In the scenario of Catalan Numbers, this principle is used to understand why dividing the count of certain paths by an integer yields the correct number of Dyck Paths or Catalan Paths. The Quotient Principle helps partition a larger set of objects into subsets that can be counted more easily.
This method offers clarity and intuition in deriving formulas by breaking down or grouping complex structures into simpler, more understandable parts.