  An antichain $\mathcal{A}$ in a poset $\mathcal{P}$ is a subset of
$\mathcal{P}$ in which no two elements are comparable. Sperner showed that the
maximal antichain in the Boolean lattice, $\mathcal{B}_n = \left\{ 0 < 1
\right\}^n$, is the largest rank (of size $\binom{n}{\lfloor n/2 \rfloor}$).
This type of problem has been since generalized, and a graded poset
$\mathcal{P}$ is said to be Sperner if the largest rank of $\mathcal{P}$ is its
maximal antichain. In this paper, we will show that the symmetric group $S_n$,
partially ordered by refinement (or equivalently by absolute order), is
Sperner.
