Burnside / Pólya / Group Actions Ladder¶
Who This Is For¶
Use this ladder when direct counting is no longer the hard part, and the real difficulty is that several representations become the same object after applying symmetries.
Warm-Up¶
- raw counting on the representation set
- one explicit symmetry group you can list by hand
Core¶
- Burnside's lemma
- counting fixed colorings under each group element
- cyclic rotations on one necklace
Stretch¶
- quarter-turn grid rotations
- reflections / dihedral cases
- cycle-index / Pólya inventory viewpoints
Example Notes¶
Exit Criteria¶
You are ready to move on when you can:
- name the representations and the symmetry group separately
- compute
fix(g)cleanly for every relevant symmetry - explain why the answer is an average of fixed counts
- tell when the starter is too narrow because reflections or richer inventories matter