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 posed in that paper, namely whether N n is a symmetric chain order for all n. This paper provides an algorithm that produces a symmetric chain decomposition (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.