Is the Symmetric Group Sperner?

TL;DR

This paper proves that the symmetric group $S_n$, ordered by refinement, has the Sperner property, meaning its largest rank forms the maximum antichain, generalizing Sperner's theorem from Boolean lattices.

Contribution

It establishes that the symmetric group $S_n$, with the refinement order, is a Sperner poset, extending Sperner's theorem to a new class of graded posets.

Findings

The symmetric group $S_n$ is Sperner under the refinement order.

Largest rank in $S_n$ forms the maximum antichain.

Generalizes Sperner's theorem to the symmetric group.

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 $\binom{n}{\lfloor n/2 \rfloor}$). 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.