Discrete Mathematics - Vol 2 | 21. Catalan Numbers - Derivation of Closed Form Formula by Abraham | Learn Smarter
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21. Catalan Numbers - Derivation of Closed Form Formula

The lecture focuses on deriving a closed-form formula for Catalan numbers, detailing the relationship between valid strings of parentheses and sequences of 1s and -1s. A reflection method is introduced to count sequences that violate partial sum conditions, leading to the calculation of the nth Catalan number. The discussion highlights the interplay between combinatorial structures and their properties.

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Sections

  • 21.1

    Catalan Numbers - Derivation Of Closed Form Formula

    This section covers the derivation of a closed-form formula for Catalan numbers through understanding sequences consisting of pairs of 1s and -1s.

    Learning Practice

  • 21.1.1

    Recap Of Previous Lecture

    This section provides a recap of problems related to Catalan numbers discussed in the previous lecture, focusing on the derivation of their closed-form formula.

    Learning Practice

  • 21.1.2

    First Problem: Strings Of Parentheses

    This section discusses the derivation of the closed-form formula for Catalan numbers using the example of valid strings of parentheses.

    Learning Practice

  • 21.1.3

    Second Problem: Sequences Of 1s And -1s

    This section discusses the number of sequences consisting of n 1s and n -1s with the constraint that the partial sum must be greater than or equal to zero.

    Learning Practice

  • 21.1.4

    Proof Strategy

    The section discusses the proof strategy for deriving a closed-form formula for Catalan numbers by analyzing sequences of 1s and -1s.

    Learning Practice

  • 21.1.5

    Cardinality Of Set A

    The section explains the cardinality of the set of sequences consisting of equal numbers of 1s and -1s, focusing on deriving the Catalan number's closed formula.

    Learning Practice

  • 21.1.6

    Cardinality Of Set B

    This section covers the derivation of the closed form formula for Catalan numbers, specifically how to find the cardinality of sequences of 1s and -1s under certain restrictions.

    Learning Practice

  • 21.1.7

    Definition Of Bad Sequence

    This section defines bad sequences in the context of Catalan numbers and outlines their properties and significance.

    Learning Practice

  • 21.1.8

    Reflection Method

    This section discusses the reflection method used to derive the closed-form formula for Catalan numbers.

    Learning Practice

  • 21.1.9

    Claims About Bad Sequence

    This section derives the closed form formula for Catalan numbers by analyzing valid and invalid sequences of 1s and -1s.

    Learning Practice

  • 21.1.10

    Constructing Sequence S'

    The section discusses how to construct a sequence S' from the set of sequences consisting of n ones and n negative ones, detailing key insights into counting and restrictions involving Catalan numbers.

    Learning Practice

  • 21.1.11

    General Process Of S' Construction

    This section delves into the derivation of Catalan numbers and the reflection method used to count sequences of numbers accurately.

    Learning Practice

  • 21.1.12

    Counting 1s And -1s In S'

    This section covers the derivation of a closed form formula for Catalan numbers, specifically through counting sequences of 1s and -1s.

    Learning Practice

  • 21.1.13

    Injective Mapping Proof

    This section explores the closed form formula for Catalan numbers using an injective mapping approach.

    Learning Practice

  • 21.1.14

    Surjective Mapping Proof

    This section explores the proof of surjective mapping in the context of Catalan numbers, illustrating the relationship between sequences of 1s and -1s.

    Learning Practice

  • 21.1.15

    Summary Of Cardinality

    This section explains the concept of cardinality in the context of Catalan numbers and their derivation through sequences of 1s and -1s.

    Learning Practice

  • 21.1.16

    Conclusion

    The conclusion summarizes the derivation of the closed form formula for Catalan numbers and highlights the introduction of the reflection method.

    Learning Practice

References

ch42.pdf

Class Notes

Memorization

What we have learnt

  • The derivation of a closed ...
  • The application of the refl...
  • The relationship between se...

Final Test

Revision Tests