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Catalan Numbers

Noor DoukmakApril 25, 2013

+Outline

Introduction

Descriptions

Generating Functions, closed-form formula

Asymptotics

Formal Grammar

Continued Fractions

Dyck Triangle

+Introduction

Belgian mathematician Eugène Charles Catalan (1814–1894)

How to bracket n+1 items using n pairs of brackets?

Cannot close a bracket without opening the corresponding one

+Small n

1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796

+Dyck Paths

+Recursive Formula?

• A structure with (n+1) pairs of brackets can be thought of as

(…)…

cn+1

+Generating Function

Let

+Generating Function

+Formula for cn

+Asymptotic Behavior

+Formal grammar

It is helpful to define production rules for words that make up the Dyck language.

Every word is either the empty word, or Dyck word sandwiched between a and b + Dyck

word

We represent this as:

+Generating Function

Note

follows from the production rules.

Let

+Generating Function

+Continued Fraction

+Stabilizes to GF

+Dyck Triangle

+Multi-variable GF

+Thank you.