catalan number presentation2
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Catalan Numbers and
Their Interpretations
Presented byPai Sukanya Suksak
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What are the Catalan Numbers?
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Historical Information
In 1730 – Chinese mathematician Antu Ming
In 1751 – Swiss mathematician Leonhard Euler
In 1838 – Belgian mathematician Eugene Charles Catalan
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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
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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
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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
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The Catalan Numbers: The Formula
Theorem For any integer n 1, the Catalan number Cn is given in term of binomial coefficients by
For n ≥ 0
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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)!
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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
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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♯
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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
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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)
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The Catalan Numbers: Some Interpretations
Then, we are going to label each branch of the tree-diagram with r and l
(Cofman, 1997)
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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)
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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)
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What is the Catalan Numbers?
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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.
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Thank you for your attention