Combinatorics

Permutations and Factorial Explosion

Count arrangements and explore how factorial growth dominates combinatorial possibilities.

Combinations, Pascal, & Sierpinski

Link binomial coefficients, Pascal’s triangle, and fractal structure in counting.

Stars and Bars

Solve partition problems using the stars-and-bars counting technique.

// Unit 2: Advanced Counting and Sieves

Inclusion-Exclusion

Use inclusion-exclusion to count overlapping sets accurately.

Derangements and 1_over_e

Study derangements and their surprising connection to e.

Pigeonhole & Birthday

Apply the pigeonhole principle to classic combinatorial arguments and paradoxes.

// Unit 3: Recurrence and Generating Functions

Catalan Numbers & DP

Generate Catalan sequences and solve recurrence relations with dynamic programming.

Generating Functions

Use generating functions as a powerful algebraic tool for counting sequences.

Integer Partitions

Explore partitions of integers and the combinatorics of additive structure.

// Unit 4: Graph Theory (Combinatorial Structures)

Trees and Cayley

Count trees and apply Cayley’s formula to spanning tree enumeration.

Graph Coloring

Study coloring problems and chromatic numbers with computational examples.

Eulerian Paths

Analyze Eulerian circuits and trails through graph traversal exercises.

// Unit 5: Extremal and Symmetrical Combinatorics

Ramsey Party

Explore Ramsey theory with party problem examples and unavoidable patterns.

Magic Squares

Investigate symmetry and constraints in magic square constructions.

Burnside’s Lemma

Use Burnside’s lemma to count orbits under group actions.