Catalan Numbers and

Their Interpretations

Presented byPai Sukanya Suksak

What are the Catalan Numbers?

Historical Information

In 1730 – Chinese mathematician Antu Ming

In 1751 – Swiss mathematician Leonhard Euler

In 1838 – Belgian mathematician Eugene Charles Catalan

General Information

Where can we find the Catalan

numbers?

Combinatorial Interpretations

Triangulation polygon

Tree Diagram

Dyck Words

Algebraic Interpretations

Dimension of Vector Space

Metrix Space

The Catalan Numbers: Sequence of Integer

Figure 1 The nth central binomial coefficient of Pascal’s triangle which is

Let’s consider the numbers in circles

n = 0

n = 1

n = 2

n = 3

n = 4

Then, if we divide each central binomial coefficient (1, 2, 6, 20, 70,…) by n + 1. (i.e., 1, 2, 3, 4, 5,… respectively)

The first fifth terms: 1, 1, 2, 5, and 14

The first fifth terms of the Catalan numbers

The Catalan Numbers: Sequence of Integer

The Catalan Numbers: The Formula

Theorem For any integer n 0, the Catalan number Cn is given in term of binomial coefficients by 

For n ≥ 0

Proof by Triangulation Definition

The Catalan Numbers: Proof of the Formula

Claim The formula of the Catalan numbers,

derives from Euler’s formula of triangulation

For n ≥ 0

Koshy first claim the triangulation formula of Euler which is

An =2 6 10 (4n − 10) n ≥ 3 (n − 1)!

Proof by Triangulation Definition

The Catalan Numbers: Proof of the Formula

Claim The formula of the Catalan numbers,

derives from Euler’s formula of triangulation

For n ≥ 0

An =2 6 10 (4n − 10) n ≥ 3 (n − 1)!

By extending the formula to include the case n = 0, 1, and 2, and rewriting the formula, Cn can be expressed as

Cn =(4n − 2) n ≥ 1 (n − 1)! Cn-1

and C0 = 1

The Catalan Numbers: Proof of the Formula

Cn =(4n − 2) (n − 1)!

Cn-1

To show the previously recursive formula is exactly the formula of Catalan numbers, Koshy applies algebraic processes as following.

(4n − 2)(4n – 6)(4n – 10)

62 (n − 1)! C0

=

= 1 (n + 1) ( )2n

n♯

The Catalan Numbers: Some Interpretations

Judita Cofman draws the relationship among combinatorial interpretations of the Catalan numbers as following.

1. The nth Catalan number is the number ofddddddddd d ddd dd ddddddddddd ddddddd d ddd d

+ 2 .

Pentagon (n = 3), C3 = 5.

http://www.toulouse.ca/EdgeGuarding/MobileGuards.html

The Catalan Numbers: Some Interpretations

Then, we are going to construct the tree-diagrams, corresponding to the partitions.

2. The nth Catalan number is the number ofddddddddd dddd dd ddddddddd dddd-ddddddd

dddd 1n + leave.

(Cofman, 1997)

The Catalan Numbers: Some Interpretations

Then, we are going to label each branch of the tree-diagram with r and l

(Cofman, 1997)

The Catalan Numbers: Some Interpretations

The codes, derived from the labelled tree-diagrams, can be formed the Dyck words of length 6

3. The nth Catalan number is the number of different ways to arrange Dyck words of length

2n.

5There are different arrangemeddd dd

Dyck words of length 2n, where =3 .

(Cofman, 1997)

The Catalan Numbers: Some Interpretations

Let r stand for “moving right” and l stand for “going up”. We will contruct monotonic paths in 3×3 square grid.

4. The nth Catalan number is the number of di ff er ent monot oni c pat hs al ong n×d ddddd

egr i d.

(Cofman, 1997)

What is the Catalan Numbers?

ReferencesThe Catalan Numbers

Conway, J., Guy, R. (1996). “THE BOOK OF NUMBERS”. Copericus, New York. Cofman, Judita (08/01/1997). "Catalan Numbers for the Classroom?". Elemente der Mathematik (0013-6018), 52 (3), p. 108.Koshy, T. (2007). “ELEMENTARY NUMBER THEORY WITH APPLICATIONS”. Boston Academic Press, Massachusetts. Stanley, R. (1944). “ENUMERATIVE COMBENATORICS”. Wadsworth & Brooks/Cole Advanced Books & Software, California.Wikipedia, “Catalan Numbers”. Retrived November 3, 2010.

Thank you for your attention