# + Catalan Numbers Noor Doukmak April 25, 2013. + Outline Introduction Descriptions Generating Functions, closed-form formula Asymptotics Formal Grammar.

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+ Catalan Numbers Noor Doukmak April 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 (…)… c n+1

+ Generating Function Let

+ Generating Function

+ Formula for c n

+ 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.

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