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

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Presentation on theme: "+ Catalan Numbers Noor Doukmak April 25, 2013. + Outline Introduction Descriptions Generating Functions, closed-form formula Asymptotics Formal Grammar."— Presentation transcript:

1 + Catalan Numbers Noor Doukmak April 25, 2013

2 + Outline Introduction Descriptions Generating Functions, closed-form formula Asymptotics Formal Grammar Continued Fractions Dyck Triangle

3 + 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

4 + Small n 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796

5 + Dyck Paths

6 + Recursive Formula? A structure with (n+1) pairs of brackets can be thought of as (…)… c n+1

7 + Generating Function Let

8 + Generating Function

9 + Formula for c n

10 + Asymptotic Behavior

11 + 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:

12 + Generating Function Note follows from the production rules. Let

13 + Generating Function

14 + Continued Fraction

15 + Stabilizes to GF

16 + Dyck Triangle

17 + Multi-variable GF

18 + Thank you.


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