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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What we have learnt
- The Catalan numbers represent the number of ways to parenthesize n + 1 numbers.
- A recurrence relation for Catalan numbers can be formulated based on splitting problems into smaller instances.
- Catalan numbers also count the valid strings of parentheses and specific sequences of 1s and -1s.
Key Concepts
- -- Catalan Numbers
- A sequence of natural numbers that occur in various counting problems, often related to recursive structures.
- -- Recurrence Relation
- An equation that defines a sequence based on previous terms, used to express the nth Catalan number.
- -- Bijection
- A one-to-one correspondence between two sets, showing that they have the same cardinality.
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