The symmetric group, ordered by refinement of cycles, is strongly Sperner

Abstract

Given a poset $(P,\leq )$, an antichain is a subset of pairwise incomparable elements of $P$. Let $(P,w)$ be a graded, weighted poset. If the maximum weight of an antichain of $P$ is equal to the weight of the largest rank of $P$, then $P$ is said to be Sperner. In 1967, Rota conjectured that the poset of partitions, ordered by refinement of blocks, is Sperner; this conjecture was later disproved by Canfield. In this paper, we consider a generalization of Rota’s conjecture and show that $S_n$, partially ordered by refinement of cycles, is strongly Sperner.