The size of the largest antichain in the partition lattice
Let \(\Pi_n\) be the poset of partitions of an \(n\)-element set, ordered by refinement, and let \(d(\Pi_n)\) be the size of the largest antichain in \(\Pi_n\). Moreover, let \(b(\Pi_n)\) be the size of the largest level in \(\Pi_n\), i.e. the largest Stirling number of the second kind where \(n\) is fixed and \(k\) varies. Let \(a:= 1/2-(e\log 2)/4\). The author proves that \(d(\Pi_n)/b(\Pi_n)\leq c\cdot n^a\cdot(\log n)^{-a-1/4}\) for a suitable constant \(c\) and \(n>1\). By a result of \textit{E. R. Canfield} and \textit{L. H. Harper} [Large antichains in the partition lattice, Random Struct. Algorithms 6, No. 1, 89-104 (1995; Zbl 0813.06002)] this upper bound is the best possible up to a multiplicative factor \(O(1)\). The main idea in the proof is the definition of an appropriate suborder (on the same basic set) for which the size of an antichain can be estimated using results from Sperner theory.
scientific article; zbMATH DE number 1212977
scientific article