feat: add Catalan numbers implementation using dynamic programming

[AliAlimohammadi]

Description

This PR adds an implementation of Catalan Numbers using dynamic programming.

The Catalan numbers are a fundamental sequence in combinatorics that appear in numerous counting problems, including:

• Counting valid parentheses combinations (e.g., ()() vs ())()
• Counting different ways to fully parenthesize expressions
• Counting full binary trees with a given number of leaves
• Counting Dyck words

Algorithm Overview:
The implementation uses dynamic programming with the recurrence relation:

• Base cases: C(0) = C(1) = 1
• Recurrence: C(n) = Σ(C(i) × C(n-i-1)) for i from 0 to n-1

Time Complexity: O(n²)
Space Complexity: O(n)

The function computes all Catalan numbers from C(0) through C(upper_limit) and returns them as a vector.

References:

• Brilliant.org - Catalan Numbers
• Wikipedia - Catalan Number

Type of change

• New feature (non-breaking change which adds functionality)

Checklist:

• I ran below commands using the latest version of rust nightly.
• I ran cargo clippy --all -- -D warnings just before my last commit and fixed any issue that was found.
• I ran cargo fmt just before my last commit.
• I ran cargo test just before my last commit and all tests passed.
• I added my algorithm to the corresponding mod.rs file within its own folder, and in any parent folder(s).
• I added my algorithm to DIRECTORY.md with the correct link.
• I checked CONTRIBUTING.md and my code follows its guidelines.

Note: All tests run quickly (well under 300ms) as they only compute Catalan numbers up to C(10).

[codecov-commenter]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 95.48%. Comparing base (eaf895d) to head (be79f44).
⚠️ Report is 1 commits behind head on master.

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #961      +/-   ##
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+ Coverage   95.40%   95.48%   +0.08%     
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  Files         334      336       +2     
  Lines       21576    21836     +260     
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+ Hits        20585    20851     +266     
+ Misses        991      985       -6     

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[AliAlimohammadi]

@siriak, This PR is ready to be merged.

[Copilot]

Pull request overview

This PR adds a new dynamic programming implementation for computing Catalan numbers, a fundamental sequence in combinatorics used for counting problems such as valid parentheses combinations, binary trees, and Dyck words. The implementation uses the standard recurrence relation with O(n²) time complexity and O(n) space complexity.

Key Changes:

• Added catalan_numbers.rs with a well-documented dynamic programming implementation
• Updated module exports in mod.rs to include the new function
• Added entry to DIRECTORY.md for documentation

Reviewed changes

Copilot reviewed 3 out of 3 changed files in this pull request and generated 3 comments.

File	Description
src/dynamic_programming/catalan_numbers.rs	New implementation of Catalan numbers using dynamic programming with comprehensive tests and documentation
src/dynamic_programming/mod.rs	Added module declaration and public export for catalan_numbers, maintaining alphabetical order
DIRECTORY.md	Added entry for Catalan Numbers algorithm (indentation issue noted)

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[AliAlimohammadi]

Review completed.

[siriak]

Looks good, thanks!