Is the Symmetric Group Sperner?
Published 2019-01-01Version 1
An antichain A in a poset P is a subset of P in which no two elements are comparable. Sperner showed that the maximal antichain in the Boolean lattice, B_n, is the largest rank (of size n choose n/2). This type of problem has since been generalized, and a graded poset P is said to be Sperner if the largest rank of 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.