The Size of the Largest Antichain in the Partition Lattice

Consider the posetΠnof partitions of ann-element set, ordered by refinement. The sizes of the various ranks within this poset are the Stirling numbers of the second kind. Leta=12−e log(2)/4. We prove the following upper bound for the ratio of the size of the largest antichain to the size of the largest rank:d(Πn ⩽)S(n, Kn)⩽c2na(log n)−a−1/4,for suitable constantc2andn>1. This upper bound exceeds the best known lower bound for the latter ratio by a multiplicative factor ofO(1).