Threshold functions for distinct parts: revisiting Erdos-Lehner

TL;DR

This paper investigates threshold functions for the probability that no two boxes have the same number of balls in various distinguishability scenarios, connecting to Erdős-Lehner partition asymptotics, motivated by statistical experiments with limited outcome information.

Contribution

It revisits and extends Erdős-Lehner results by analyzing threshold functions in partition problems with distinguishable and non-distinguishable balls and boxes.

Findings

Derived threshold functions for different distinguishability cases.

Connected partition asymptotics to probabilistic threshold phenomena.

Provided insights relevant to experiments with limited outcome distinguishability.

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.