On a continuous analog of Sperner's problem
Let the set of partitions of an n-element set S be partially ordered by refinement, and define the rank of a partition of S to be the number of nonempty subsets in it, so that the number of partitions of rank k is the Stirling number S(n,k) of the second kind. It was conjectured by Rota that the size of the largest antichain of partitions of S is \(\max_{k}S(n,k)\). However, \textit{R. Canfield} [Advances Math. 29, 1-10 (1978; Zbl 0378.05003)] showed that if n is sufficiently large then the conjecture is false: larger antichains exist. This raised the following question: is the ratio of the size of the largest antichain to \(\max_{k}S(n,k)\) asymptotic to 1? By using arguments based on the assumption that the lattice of partitions may be approximated by a Gaussian process, the author shows that this asymptotic conjecture is likely to be false, there existing an antichain whose size is at least 1.69 times the largest Stirling number.
scientific article; zbMATH DE number 3943830