Is the Symmetric Group Sperner?

Abstract

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 $\left(\genfrac{}{}{0ex}{}{n}{\lfloor n/2\rfloor }\right)$). 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.

7 pages, 7 figures