+|
+
+**Computer Science Fundamentals**
+- ๐๏ธ Data Structures (Arrays, Trees, Graphs, Heaps)
+- ๐ Sorting (Quick, Merge, Heap, Radix, etc.)
+- ๐ Searching (Binary, Ternary, Jump, etc.)
+- ๐ Backtracking (N-Queens, Sudoku, Maze)
+- ๐ Divide and Conquer
+- ๐ฐ Dynamic Programming
+- ๐ฏ Greedy Algorithms
+- ๐ Graph Algorithms (BFS, DFS, Dijkstra, Floyd-Warshall)
+
+**Mathematics & Science**
+- โ Mathematical Algorithms
+- ๐ข Number Theory
+- ๐ Geometry & Computational Geometry
+- ๐งฎ Linear Algebra & Matrix Operations
+- ๐ Statistics & Probability
+- โ๏ธ Physics Simulations
+- ๐ Geodesy
+
+ |
+
+
+**Machine Learning & AI**
+- ๐ค Machine Learning (Regression, Classification)
+- ๐ง Neural Networks
+- ๐งฌ Genetic Algorithms
+- ๐ฒ Fuzzy Logic
+- ๐ Data Mining
+
+**Applied Computer Science**
+- ๐ Cryptography & Ciphers
+- ๐ผ๏ธ Computer Vision
+- ๐จ Digital Image Processing
+- ๐ต Audio Filters
+- ๐ Web Programming
+- ๐ก Networking & Data Transfer
+- ๐๏ธ Data Compression
+- ๐น Financial Algorithms
+- โก Electronics & Circuit Design
+
+ |
+
+
From 3a6c6f68c6f067ee1ce5c95945f3107d1990f878 Mon Sep 17 00:00:00 2001
From: Divyansh Raj Soni <110761086+DivyanshRajSoni@users.noreply.github.com>
Date: Mon, 6 Oct 2025 12:10:57 +0530
Subject: [PATCH 3/4] Add Catalan number algorithm with three implementations
Implements Catalan number calculation using:
- Dynamic programming approach
- Recursive with memoization
- Binomial coefficient formula
Includes 18 passing doctests, type hints, and comprehensive documentation.
---
maths/catalan_number.py | 175 ++++++++++++++++++++++++++++++++++++++++
1 file changed, 175 insertions(+)
create mode 100644 maths/catalan_number.py
diff --git a/maths/catalan_number.py b/maths/catalan_number.py
new file mode 100644
index 000000000000..d0b82184ae8e
--- /dev/null
+++ b/maths/catalan_number.py
@@ -0,0 +1,175 @@
+"""
+Catalan Numbers
+
+Catalan numbers form a sequence of natural numbers that occur in various counting
+problems in combinatorics. The nth Catalan number can be expressed directly in
+terms of binomial coefficients.
+
+Formula: C(n) = (2n)! / ((n + 1)! * n!)
+
+Alternative formula: C(n) = C(n-1) * 2(2n-1) / (n+1)
+
+Applications:
+- Number of different ways n + 1 factors can be completely parenthesized
+- Number of different binary search trees with n keys
+- Number of paths with n steps east and n steps north that don't cross diagonal
+- Number of ways to triangulate a polygon with n + 2 sides
+
+Reference: https://en.wikipedia.org/wiki/Catalan_number
+"""
+
+
+def catalan_number(node_count: int) -> int:
+ """
+ Calculate the nth Catalan number using dynamic programming approach.
+
+ Args:
+ node_count: A non-negative integer representing the position in sequence
+
+ Returns:
+ The nth Catalan number
+
+ Raises:
+ ValueError: If node_count is negative
+
+ Examples:
+ >>> catalan_number(0)
+ 1
+ >>> catalan_number(1)
+ 1
+ >>> catalan_number(5)
+ 42
+ >>> catalan_number(10)
+ 16796
+ >>> catalan_number(15)
+ 9694845
+ >>> catalan_number(-1)
+ Traceback (most recent call last):
+ ...
+ ValueError: node_count must be a non-negative integer
+ >>> catalan_number(3.5)
+ Traceback (most recent call last):
+ ...
+ ValueError: node_count must be a non-negative integer
+ """
+ if not isinstance(node_count, int) or node_count < 0:
+ raise ValueError("node_count must be a non-negative integer")
+
+ if node_count <= 1:
+ return 1
+
+ # Dynamic programming approach
+ catalan = [0] * (node_count + 1)
+ catalan[0] = catalan[1] = 1
+
+ for i in range(2, node_count + 1):
+ for j in range(i):
+ catalan[i] += catalan[j] * catalan[i - 1 - j]
+
+ return catalan[node_count]
+
+
+def catalan_number_recursive(node_count: int) -> int:
+ """
+ Calculate the nth Catalan number using recursive formula with memoization.
+
+ Args:
+ node_count: A non-negative integer representing the position in sequence
+
+ Returns:
+ The nth Catalan number
+
+ Raises:
+ ValueError: If node_count is negative
+
+ Examples:
+ >>> catalan_number_recursive(0)
+ 1
+ >>> catalan_number_recursive(1)
+ 1
+ >>> catalan_number_recursive(5)
+ 42
+ >>> catalan_number_recursive(10)
+ 16796
+ >>> catalan_number_recursive(-1)
+ Traceback (most recent call last):
+ ...
+ ValueError: node_count must be a non-negative integer
+ """
+ if not isinstance(node_count, int) or node_count < 0:
+ raise ValueError("node_count must be a non-negative integer")
+
+ memo: dict[int, int] = {}
+
+ def helper(n: int) -> int:
+ if n <= 1:
+ return 1
+ if n in memo:
+ return memo[n]
+
+ result = 0
+ for i in range(n):
+ result += helper(i) * helper(n - 1 - i)
+
+ memo[n] = result
+ return result
+
+ return helper(node_count)
+
+
+def catalan_number_binomial(node_count: int) -> int:
+ """
+ Calculate the nth Catalan number using binomial coefficient formula.
+
+ Formula: C(n) = (2n)! / ((n + 1)! * n!)
+ which equals: C(2n, n) / (n + 1)
+
+ Args:
+ node_count: A non-negative integer representing the position in sequence
+
+ Returns:
+ The nth Catalan number
+
+ Raises:
+ ValueError: If node_count is negative
+
+ Examples:
+ >>> catalan_number_binomial(0)
+ 1
+ >>> catalan_number_binomial(1)
+ 1
+ >>> catalan_number_binomial(5)
+ 42
+ >>> catalan_number_binomial(10)
+ 16796
+ >>> catalan_number_binomial(15)
+ 9694845
+ >>> catalan_number_binomial(-1)
+ Traceback (most recent call last):
+ ...
+ ValueError: node_count must be a non-negative integer
+ """
+ if not isinstance(node_count, int) or node_count < 0:
+ raise ValueError("node_count must be a non-negative integer")
+
+ if node_count <= 1:
+ return 1
+
+ # Calculate binomial coefficient C(2n, n)
+ result = 1
+ for i in range(node_count):
+ result = result * (2 * node_count - i) // (i + 1)
+
+ # Divide by (n + 1)
+ return result // (node_count + 1)
+
+
+if __name__ == "__main__":
+ import doctest
+
+ doctest.testmod()
+
+ # Print first 15 Catalan numbers
+ print("First 15 Catalan numbers:")
+ for i in range(15):
+ print(f"C({i}) = {catalan_number(i)}")
From 374b63fa7d5db04d80ae0d81caf9bd10b9ef7dfa Mon Sep 17 00:00:00 2001
From: "pre-commit-ci[bot]"
<66853113+pre-commit-ci[bot]@users.noreply.github.com>
Date: Mon, 6 Oct 2025 06:53:51 +0000
Subject: [PATCH 4/4] [pre-commit.ci] auto fixes from pre-commit.com hooks
for more information, see https://pre-commit.ci
---
README.md | 2 +-
1 file changed, 1 insertion(+), 1 deletion(-)
diff --git a/README.md b/README.md
index 9ec57e4a28df..4cf4672756e9 100644
--- a/README.md
+++ b/README.md
@@ -1,4 +1,4 @@
-