The size of the largest antichain in the partition lattice

摘要

Consider the poset Pi(n) of partitions of an n-element set, ordered by refinement. The sizes of the various ranks within this poset are the Stirling numbers of the second kind. Let a = 1/2 - 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(Pi(n), less than or equal to)/S(n, K-n)less than or equal to c(2)n(a)(log n)(-a-1/4), for suitable constant c(2) and n > 1. This upper bound exceeds the best known lower bound for the latter ratio by a multiplicative factor of O(1). (C) 1998 Academic Press. [References: 21]