Is Canfield Right? On the Asymptotic Coefficients for the Maximum Antichain of Partitions and Related Counting Inequalities

This paper dates back to the asymptotic solutions of Rota’s problem on the size of maximum antichain in the set partition lattice by Canfield and Harper and others. The knowledge of asymptotic coefficients could pave the way to the asymptotic solutions of such problems as (maximal) antichain counting in partition lattices. In addition to our attempt to reduce uncertainty in the values of these coefficients, we provide some inequalities for the discrepancy between the number of antichains and maximal antichains in partition lattices and give alternative proof for the number of maximal antichains obtained by us recently and recorded in the Online Encyclopaedia of Integer Sequences ( https://oeis.org/A358041 ).