Necklaces: distinct circular arrangements of n beads in k colors, up to rotation. Count: M(n,k)=(1/n)Σ_{d|n}φ(d)·k^{n/d}. Bracelets add reflection symmetry. Uses Burnside's lemma and Euler's totient function.
How to Use
Enter n beads and k colors:
- Necklaces: Rotations only
- Bracelets: +reflections
- Formula: Burnside breakdown
Burnside's Lemma
Count orbits = (1/|G|)Σ_{g∈G}|Fix(g)|. For rotations: rotation by d positions fixes colorings periodic with period gcd(d,n). The formula simplifies to (1/n)Σ_{d|n}φ(d)·k^{n/d}.
Applications
Chemistry (cyclic molecules), music theory (pitch class sets), combinatorial design, coding theory (cyclic codes), and DNA sequence analysis all use necklace counting.
Step-by-Step Instructions
- 1Enter bead count.
- 2Enter color count.
- 3Compute necklaces.
- 4Compute bracelets.
- 5View formula.