Necklaces with k colors on n beads, ordered bead-by-bead: the quotient [k−1]n/G of a product of chains
colors
group
n
layout
orbit size 1n
Made for Math 372: Combinatorics at Reed College, fall 2026. Elements are orbits of the cyclic group Cn (or the dihedral group Dn) acting on words in {0,…,k−1}n, i.e. necklaces (or bracelets); the order is induced from the product of chains, generalizing the quotients Bn/G of Chapter 5 of Stanley's Algebraic Combinatorics (k = 2 is exactly Bn/Cn). Part of e-infinity.space/viz.