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Number Properties: Help & Review8 chapters | 55 lessons

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

This lesson will define the Catalan numbers. We will then review the history of these numbers, including some of the mathematicians and applications that led to the discovery and development of these numbers.

The **Catalan numbers** are a fascinating sequence of numbers in mathematics that show up in many different applications. Technically speaking, the *n*th Catalan number, *C**n*, is given by the following formula:

*C**n*= (2*n*)! / ((*n*+ 1)!*n*!), where*n*! =*n*⋅ (*n*- 1) ⋅ (*n*- 2) ⋅ … ⋅ 3 ⋅ 2 ⋅ 1

For example, the 10th Catalan number would be found by plugging 10 into the formula for *n*.

We get that *C*10 = 16796.

Well, sure these numbers sound neat, but we described them as 'fascinating'. Don't worry - they are! You see, these numbers show up in a large variety of applications in combinatorics, or counting processes. For instance, consider the set of integers from 1 to *n*, or the first *n* integers.

- {1, 2, 3, …,
*n*}

The number of permutations, or orderings, of this set, such that there are no three consecutive integers anywhere in the permutation is equal to the *n*th Catalan number, *C**n*. To illustrate this, consider the following set:

- {1, 2, 3}

The permutations that don't include three consecutive integers of this set are the permutations that don't include 123, and those are as follows:

- 132, 213, 231, 312, 321

We see there are 5 of them. Since the set, {1, 2, 3} has 3 integers in it, the number of permutations of the set that don't include three consecutive integers should be equal to the third Catalan number, or *C*3. Therefore, *C*3 should equal five if we've got them all in the list above.

Sure enough, *C*3 = 5, so there are 5 permutations of the set {1, 2, 3}, such that none of the permutations contain three consecutive integers.

The history of the discovery and development of the Catalan numbers is just as interesting as the numbers themselves, and the history revolves around a few of the applications of these numbers. Let's take a look at what makes the Catalan numbers so fascinating!

There are three mathematicians that deserve special mention when it comes to the discovery and development of the Catalan numbers: Leohnard Euler, Eugene Charles Catalan, and Sharabiin Myangat.

Leonhard Euler is considered to be the first person to discover the Catalan numbers. In 1751, there was a series of correspondence between Euler and a German mathematician named Christian Goldbach, in which Euler was trying to figure out how many ways to split a convex polygon into triangles by connecting its vertices with line segments so that none of the line segments cross. It turns out that if a polygon has *n* + 2 sides, then the number of ways to do this is equal to the *C**n*, or the *n*th Catalan number.

Hmmm, so if Euler was the first to discover the Catalan numbers, then why aren't they called the Euler numbers?

Well, for one thing, Euler has a lot of numbers named after him, so that name was already taken. However, for a time, the Catalan numbers were actually called the Euler-Segner numbers. This was due to the work with these numbers that Euler and another mathematician, Johann Andres Von Segner, performed in the 18th century.

The real reason for the Catalan number's name is that it was Eugene Charles Catalan, a French and Belgian mathematician, who was the first to actually describe these numbers as a sequence and give a well-defined formula for the *n*th Catalan number.

Eugene Charles Catalan came up with the sequence and formula for Catalan numbers when he was studying groupings using parentheses. To group parentheses properly, each open parenthesis must have a corresponding closed parenthesis. For example, (( )) or ()() are both proper groupings with two pairs of parentheses, but (( ) is not, because there is no corresponding closed parenthesis for one of the open ones.

Surprise, surprise! This is another application of the Catalan numbers! The number of groupings there are to group *n* pairs of parenthesis is equal to *C**n*, or the *n*th Catalan number.

Pretty neat, huh?

There is one more mathematician that deserves mention when it comes to the Catalan numbers, and that is Sharabiin Myangat, also called Minggatu or Ming Antu. Myangat was a Mongolian mathematician that wrote the book 'Quick Methods for Accurate Values of Circle Segments'. In this book, he used the Catalan numbers in expressing series expansions of sin(*ma*) for specific values of *m*.

This was in the 1730's, and these numbers were not yet well-known as a specific sequence of numbers. Therefore, Myangat is credited with introducing the Catalan numbers to China in the 1730's, but they were not yet named the Catalan numbers at that point.

The **Catalan numbers** are a sequence of numbers in mathematics. The *n*th Catalan number, *C**n*, is given by the following formula:

*C**n*= (2*n*)! / ((*n*+ 1)!*n*!), where*n*! =*n*⋅ (*n*- 1) ⋅ (*n*- 2) ⋅ … ⋅ 3 ⋅ 2 ⋅ 1

These numbers show up in a huge number of applications in **combinatorics**, and it was these applications that brought about their discovery.

Though these numbers have been used and studied by many mathematicians through the years, there are three mathematicians that are most well-known for their part in the history of the Catalan numbers, and they are Leonhard Euler, Eugene Charles Catalan, and Sharabiin Myangat. Each of these mathematicians played an important part in discovering applications that involve the Catalan numbers along with the definition and formula for the Catalan numbers themselves.

These Catalan numbers really are fascinating!

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Number Properties: Help & Review8 chapters | 55 lessons

- What are the Different Types of Numbers? 6:56
- What Is The Order of Operations in Math? - Definition & Examples 5:50
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