On a continuous analog of Sperner's problem
type: Item
instance of: scholarly article
title: On a continuous analog of Sperner's problem
zbMATH Open document ID: 0588.05007
published in: Pacific Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.pjm/1102706449
review text: 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.
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 06A06
zbMATH DE Number: 3943830
zbMATH Keywords: partitions
zbMATH Keywords: antichain
author: Lawrence H. Harper
MaRDI profile type: Publication
OpenAlex ID: W1963871572
DOI: 10.2140/PJM.1985.118.411
description: scientific article; zbMATH DE number 3943830