Discrete Mathematics - Vol 2 | 20. Catalan Numbers by Abraham | Learn Smarter
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20. Catalan Numbers

The chapter explores the concept of Catalan numbers, which arise in various combinatorial problems, particularly those involving parenthesization. It discusses the formulation of a recurrence relation for the Catalan numbers and demonstrates how they can be applied to different types of counting problems, including valid parentheses strings and specific sequences of 1s and -1s. The chapter concludes with a method to find a closed-form expression for the nth Catalan number.

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Sections

  • 20.1

    Catalan Numbers

    Catalan numbers are a sequence of natural numbers that have significant applications in combinatorial problems.

    Learning Practice

  • 20.2

    Formulating The Problem

    This section discusses formulating recurrence relations for counting problems, particularly focusing on Catalan numbers.

    Learning Practice

  • 20.2.1

    Understanding C(N)

    This section explores the concept of Catalan numbers and how to calculate C(n), the number of ways to parenthesize a sequence of numbers.

    Learning Practice

  • 20.2.2

    Recurrence Equation

    This section introduces recurrence equations through the context of Catalan numbers, which count distinct ways to parenthesize products of numbers and various combinatorial structures.

    Learning Practice

  • 20.3

    Parenthesizing Orders

    This section introduces Catalan numbers through the context of counting parenthesizing orders for multiplication.

    Learning Practice

  • 20.3.1

    Final Dot Interpretation

    This section delves into the concept of Catalan numbers by exploring how to count valid parenthesizations of a set of numbers.

    Learning Practice

  • 20.4

    Valid Strings Of Parentheses

    This section focuses on the concept of Catalan numbers, specifically counting valid parentheses strings and parenthesization of products.

    Learning Practice

  • 20.4.1

    Formulation Of Valid Strings

    This section discusses the formulation of valid strings using recurrence relations, particularly focusing on the Catalan numbers.

    Learning Practice

  • 20.4.2

    Bijection Between Problems

    This section explores the concept of Catalan numbers through the formulation of systematic problem-solving patterns, particularly focusing on how specific problems relate to one another via bijections.

    Learning Practice

  • 20.5

    Finding Closed Form Of Catalan Numbers

    This section explores Catalan numbers, deriving their closed form through recurrence relations and exploring various combinatorial problems.

    Learning Practice

  • 20.5.1

    New Problem Of Sequences

    This section introduces Catalan numbers, focusing on their properties and applications through example problems like parenthesizing products and valid parenthesis strings.

    Learning Practice

  • 20.5.2

    Deriving Closed Formula

    This section introduces Catalan numbers through recurrence relations and examples of counting valid parenthetizations.

    Learning Practice

  • 20.6

    Conclusion And Summary

    This section summarizes the significance of Catalan numbers and their relation to various combinatorial problems.

    Learning Practice

References

ch41.pdf

Class Notes

Memorization

What we have learnt

  • The Catalan numbers represe...
  • A recurrence relation for C...
  • Catalan numbers also count ...

Final Test

Revision Tests