Threshold functions for distinct parts: revisiting Erdos-Lehner

Abstract: We study four problems: put $n$ distinguishable/non-distinguishable balls into $k$ non-empty distinguishable/non-distinguishable boxes randomly. What is the threshold function $k=k(n) $ to make almost sure that no two boxes contain the same number of balls? The non-distinguishable ball problems are very close to the Erd\H os--Lehner asymptotic formula for the number of partitions of the integer $n$ into $k$ parts with $k=o(n^{1/3})$. The problem is motivated by the statistics of an experiment, where we only can tell whether outcomes are identical or different.