Theoretical / proof-based
Combinatorics · Graduate · Math
Learning objectives from the Combinatorics syllabus, grouped by unit. Click an objective for study materials.
Topics typically covered
Click a topic for the full text and related unit practice.
Graduate Combinatorics — scope drawn from open graduate combinatorics notes and typical US graduate combinatorics syllabi.
Enumerative combinatorics
- Generating functions — Generating functions
- Exponential — Exponential
- Recurrence relations — Recurrence relations
- Transfer matrices — Transfer matrices
- Inclusion–exclusion — Inclusion–exclusion
- Möbius inversion on posets — Möbius inversion on posets
- Polya enumeration and cycle index — Polya enumeration and cycle index
- Asymptotic enumeration (introduction) — Asymptotic enumeration (introduction)
Graph theory and extremal problems
- Turán's theorem — Turán's theorem
- Extremal graph theory — Extremal graph theory
- Ramsey theory — Ramsey theory
- Infinite (introduction) — Infinite (introduction)
- Matchings, factors, and Tutte's theorem — Matchings, factors, and Tutte's theorem
- Planarity — Planarity
- Wagner's theorems (statements) — Wagner's theorems (statements)
- Spectral graph theory (introduction) — Spectral graph theory (introduction)
Algebraic and geometric combinatorics
- Combinatorial Nullstellensatz — Combinatorial Nullstellensatz
- Polynomial method — Polynomial method
- Matroids: definitions, bases, and duality — Matroids: definitions, bases, and duality
- Symmetric functions — Symmetric functions
- Young tableaux (introduction) — Young tableaux (introduction)
- Incidence algebras and Möbius functions — Incidence algebras and Möbius functions
- Lattice paths and binomial identities — Lattice paths and binomial identities
Learning objectives
Click an objective for study materials.
Enumerative combinatorics
- Generating functions — Generating functions
- Exponential — Exponential
- Recurrence relations — Recurrence relations
- Transfer matrices — Transfer matrices
- Inclusion–exclusion — Inclusion–exclusion
- Möbius inversion on posets — Möbius inversion on posets
- Polya enumeration and cycle index — Polya enumeration and cycle index
- Asymptotic enumeration (introduction) — Asymptotic enumeration (introduction)
Graph theory and extremal problems
- Turán's theorem — Turán's theorem
- Extremal graph theory — Extremal graph theory
- Ramsey theory — Ramsey theory
- Infinite (introduction) — Infinite (introduction)
- Matchings, factors, and Tutte's theorem — Matchings, factors, and Tutte's theorem
- Planarity — Planarity
- Wagner's theorems (statements) — Wagner's theorems (statements)
- Spectral graph theory (introduction) — Spectral graph theory (introduction)
Algebraic and geometric combinatorics
- Combinatorial Nullstellensatz — Combinatorial Nullstellensatz
- Polynomial method — Polynomial method
- Matroids: definitions, bases, and duality — Matroids: definitions, bases, and duality
- Symmetric functions — Symmetric functions
- Young tableaux (introduction) — Young tableaux (introduction)
- Incidence algebras and Möbius functions — Incidence algebras and Möbius functions
- Lattice paths and binomial identities — Lattice paths and binomial identities
Definitions and structure
- Generating functions — Generating functions
- Exponential — Exponential
- Recurrence relations — Recurrence relations
- Transfer matrices — Transfer matrices
- Inclusion–exclusion — Inclusion–exclusion
Proofs and reasoning
- Möbius inversion on posets — Möbius inversion on posets
- Polya enumeration and cycle index — Polya enumeration and cycle index
- Asymptotic enumeration (introduction) — Asymptotic enumeration (introduction)
- Turán's theorem — Turán's theorem
- Extremal graph theory — Extremal graph theory
Abstraction and generalization
- Ramsey theory — Ramsey theory
- Infinite (introduction) — Infinite (introduction)
- Matchings, factors, and Tutte's theorem — Matchings, factors, and Tutte's theorem
- Planarity — Planarity
- Wagner's theorems (statements) — Wagner's theorems (statements)
Modeling and computation
- Apply generating functions in engineering contexts — Apply generating functions in engineering contexts
- Apply exponential in engineering contexts — Apply exponential in engineering contexts
- Apply recurrence relations in engineering contexts — Apply recurrence relations in engineering contexts
- Apply transfer matrices in engineering contexts — Apply transfer matrices in engineering contexts
- Apply inclusion–exclusion in engineering contexts — Apply inclusion–exclusion in engineering contexts
Data and technology
- Use software to explore combinatorics problems numerically — Use software to explore combinatorics problems numerically
- Interpret computational results against analytic predictions — Interpret computational results against analytic predictions
- Build spreadsheets or scripts for routine calculations — Build spreadsheets or scripts for routine calculations
- Visualize functions, fields, or datasets tied to course topics — Visualize functions, fields, or datasets tied to course topics
- Connect course methods to lab, industry, or research workflows — Connect course methods to lab, industry, or research workflows
Problem-solving practice
- Möbius inversion on posets — Möbius inversion on posets
- Polya enumeration and cycle index — Polya enumeration and cycle index
- Asymptotic enumeration (introduction) — Asymptotic enumeration (introduction)
- Turán's theorem — Turán's theorem
- Extremal graph theory — Extremal graph theory
Multi-Unit Problems
Course-level sets that combine skills across study units (coming soon).
Browse Multi-Unit ProblemsWhat each unit includes
Open a unit below for full materials. Typical resources:
- Study guide
- Exam Strategy
- Common Mistakes
- Worksheets
- Word problems
- Mixed Practice
- Multi-Unit Problems
- Review
- Practice test
- Answer key
Study units
Each unit includes a study guide, worksheets, review, practice test, and answer key. One unit is free; subscribe for the full class.
- Enumerative combinatorics
Generating functions
Coming soon - Graph theory and extremal problems
Turán's theorem
Coming soon - Algebraic and geometric combinatorics
Combinatorial Nullstellensatz
Coming soon
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