feat: Add Catalan Numbers algorithm implementation to mathematics module

[iampratik13]

This PR introduces an implementation of Catalan Numbers calculation algorithms to the mathematics module. Catalan numbers are widely used in combinatorics and appear in various counting problems in computer science and mathematics.

What’s Added
• Dynamic Programming Approach (catalan_dp): Efficient O(n) time complexity implementation.
• Direct Formula Approach (catalan_formula): Alternative calculation using the combinatorial formula.
• Utility Functions:
• first_n_catalan(): Generates the first n Catalan numbers.
• catalan_applications(): Demonstrates real-world applications of Catalan numbers.

Key Features
• Performance: O(n) time and space complexity.
• Educational: Includes mathematical explanations and documentation.
• Practical: Multiple approaches provided for flexibility.
• Examples: Sample usage demonstrating various applications.

Applications Covered
• Counting valid parentheses combinations
• Enumerating full binary trees
• Triangulating convex polygons
• Counting monotonic lattice paths
• Arranging non-attacking rooks

File Added
• mathematics/catalan_numbers.r

Testing

The implementation includes several example use cases and has been validated against known Catalan number sequences.

⸻

Type: Feature
Module: Mathematics
Breaking Changes: None

[Copilot]

Pull Request Overview

This PR adds a comprehensive implementation of Catalan Numbers algorithms to the mathematics module. Catalan numbers are important in combinatorics and appear in various counting problems in computer science.

• Implements two different calculation approaches: dynamic programming and direct formula
• Provides utility functions for generating sequences and demonstrating applications
• Includes comprehensive documentation and examples for educational purposes