The necklace poset is a symmetric chain order
Let $N _ n$ denote the quotient poset of the Boolean lattice, $B _ n$, under the relation equivalence under rotation. Griggs, Killian, and Savage proved that $N _ p$ is a symmetric chain order for prime p. In this paper, we settle the question of whether this poset is a symmetric chain order for all n by providing an algorithm that produces a symmetric chain decompostion (or SCD). We accomplish this by modifying bracketing from Greene and Kleitman. This allows us to take appropriate “middles” of certain chains from the Greene-Kleitman SCD for $B _ n$. We also prove additional properties of the resulting SCD and show that this settles a related conjecture.