Difference between revisions of "Catalan sequence"

Latest revision as of 14:00, 5 July 2024

The Catalan sequence is a sequence of positive integers that arise as the solution to a wide variety of combinatorial problems. The first few terms of the Catalan sequence are $C_0 = 1$ , $C_1 = 1$ , $C_2 = 2$ , $C_3 = 5$ , .... In general, the $n$ th term of the Catalan sequence is given by the formula $C_n = \frac{1}{n + 1}\binom{2n}{n}$ , where $\binom{2n}{n}$ is the $n$ th central binomial coefficient.

Introduction

The Catalan sequence can be used to find:

The number of ways to arrange $n$ pairs of matching parentheses.
The number of ways a convex polygon of $n+2$ sides can be split into $n$ triangles by $n - 1$ nonintersection diagonals.
The number of rooted binary trees with exactly $n+1$ leaves.
The number of paths with $2n$ steps on a rectangular grid from $(0,0)$ to $(n,n)$ that do not cross above the main diagonal.

A recursive definition of the Catalan sequence is $C_n = \sum_{k=0}^{n-1} C_k C_{n-1-k}$

Example

In how many ways can the product of $n$ ordered number be calculated by pairs? For example, the possible ways for $a\cdot b\cdot c\cdot d$ are $a((bc)d), a(b(cd)), (ab)(cd), ((ab)c)d,$ and $(a(bc))d$ .

Solution

External Links

@@ Line 1: / Line 1: @@
-The '''Catalan Numbers''' are a [[sequence]] of numbers that show up in a variety of instances.
+The '''Catalan sequence''' is a [[sequence]] of [[positive integer]]s that arise as the solution to a wide variety of combinatorial problems.  The first few terms of the Catalan sequence are <math>C_0 = 1</math>, <math>C_1 = 1</math>, <math>C_2 = 2</math>, <math>C_3 = 5</math>, ....  In general, the <math>n</math>th term of the Catalan sequence is given by the formula <math>C_n = \frac{1}{n + 1}\binom{2n}{n}</math>, where <math>\binom{2n}{n}</math> is the <math>n</math>th central [[binomial coefficient]].
 == Introduction ==
-Catalan numbers can be used to find:
+The Catalan sequence can be used to find:
-# The number of ways to arrange <math>n</math> pairs of matching parentheses
+# The number of ways to arrange <math>n</math> pairs of matching parentheses.
-# The number of ways a convex polygon of <math>n+2</math> sides can be split into <math>n</math> triangles
+# The number of ways a [[convex polygon]] of <math>n+2</math> sides can be split into <math>n</math> [[triangle]]s by <math>n - 1</math> nonintersection [[diagonal]]s.
-# The number of rooted binary trees with exactly <math>n+1</math> leaves
+# The number of [[Tree (graph theory)#Rooted Tree|rooted binary tree]]s with exactly <math>n+1</math> leaves.
+# The number of paths with <math>2n</math> steps on a rectangular grid from <math>(0,0)</math> to <math>(n,n)</math> that do not cross above the main diagonal.
+A recursive definition of the Catalan sequence is <math>C_n = \sum_{k=0}^{n-1} C_k C_{n-1-k}</math>
 === Example ===
@@ Line 13: / Line 16: @@
 === Solution ===
-== See also ==
+== See Also ==
 * [[Combinatorics]]
+* [[Fibonacci sequence]]
+== External Links==
+* [http://www-math.mit.edu/~rstan/ec/catadd.pdf Richard Stanley's Catalan Compendium (.pdf)]
+* [http://www.research.att.com/~njas/sequences/A000108 Catalan numbers in the OEIS]
 [[Category:Combinatorics]]

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