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Catalan Numbers: Formula, Applications & Example

Instructor: Laura Pennington

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

Catalan numbers are an important and prevalent sequence in mathematics. In this lesson, you'll find the formula for identifying Catalan numbers and learn how to apply them through some examples.

Catalan Numbers

Jessie is an artist who's trying to figure out how many different ways she can split a pentagonal sculpture into triangles by adding beams between the pentagon's vertices (with no beams crossing) to determine how the finished sculpture will look. One way Jessie could do this is by drawing all of the possibilities.


catalan1


But how does Jessie know if her drawings represent all of the possible combinations? As it turns out, there's an easy way to do this - by using Catalan numbers! Catalan numbers are a sequence of positive integers, where the nth term in the sequence, denoted Cn, is found in the following formula:

  • Cn = (2n)! / ((n + 1)!n!)

Here n!, pronounced n factorial, is equal to the product of all of the integers from n down to 1.

  • (n) ⋅ (n - 1) ⋅ (n - 2) ⋅ … ⋅ 2 ⋅ 1

Sample Problem

For example, to find C7, plug n = 7 into the formula and simplify:


Finding C 7
catalan2


In this sample problem, C7 = 429.

If we start at n = 0, then the first ten Catalan numbers are as follows:

  • 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, …

What do these numbers have to do with Jessie's pentagonal sculpture? Let's take a look!

Application: Pentagon Problem

Catalan numbers are directly related to how many ways we can split an n-gon into triangles by connecting vertices where no two line segments cross. The number of possibilities is equal to Cn - 2.

In Jessie's case, we want to find the number of ways to organize a pentagon into triangles so that no two line segments that connect the vertices cross. A pentagon has five sides, so we need to find C5 - 2, or C3, by plugging n = 3 into the Catalan number formula:


Finding C 3
catalan3


According to this equation, there are five ways to split a pentagon into triangles by connecting the vertices with line segments that do not cross. This supports the drawings shown at the beginning of the lesson.

This isn't the only way to apply Catalan numbers. Let's consider one more way these numbers are used in mathematics.

Applications in Combinatorics

The nth Catalan number, or Cn, is also equal to the number of permutations, or orderings, of the set of integers between 1 and n, or {1, …, n}, such that none of the permutations include three consecutive integers.

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