The Size of the Largest Antichain in the Partition Lattice

Consider the poset?nof 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?elog(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(?n?)S(n,Kn)?c2na(logn)?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).

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