Combinatorics -> Burnside / Pólya / Group Actions

Count colorings up to explicit symmetry by averaging fixed representations over a group action, starting with cyclic rotations on necklaces.

• Topic slug: combinatorics/burnside-polya
• Tutorial page: Open tutorial
• Ladder page: Open ladder
• Repo problems currently tagged here: 1
• Repo companion pages: 3
• Curated external problems: 2

Microtopics

• burnside-lemma
• group-actions
• orbit-counting
• fixed-colorings
• cyclic-rotations
• cycle-index

Learning Sources

Source	Type
cp-algorithms Burnside / Pólya	Reference
Competitive Programmer's Handbook	Reference
Guide to Competitive Programming	Reference

Practice Sources

Source	Type
CSES Counting Necklaces	Practice
CSES Counting Grids	Practice

Repo Companion Material

Material	Type
Burnside / Pólya hot sheet	quick reference
Modular Arithmetic hot sheet	adjacent sheet
Counting Basics	prereq topic

Curated External Problems

Core

Problem	Source	Difficulty	Context	Style	Prerequisites	Tags	Why it fits
Counting Necklaces	CSES	Hard	Group Actions, Orbit Counting	Combinatorics; Math	Basic Counting; Modular Exponentiation; GCD	Necklaces; Rotations; Mod Arithmetic	The clean first Burnside problem: cyclic rotations on one n-cycle with a fixed-coloring count of m^gcd(n, r).

Stretch

Problem	Source	Difficulty	Context	Style	Prerequisites	Tags	Why it fits
Counting Grids	CSES	Hard	Rotation Symmetry	Combinatorics; Math	Burnside Lemma; Modular Exponentiation	Grids; Quarter Turns; Mod Arithmetic	The next rep after necklaces: same orbit-counting worldview, but each rotation creates a different cell-cycle structure.

Repo Problems

Code	Title	Fit	Difficulty	Pattern	Note	Solution
COUNTINGNECKLACES	Counting Necklaces	primary	hard	burnside lemma; rotation symmetry; orbit counting	Note	Code

Regeneration

python3 scripts/generate_problem_catalog.py