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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