Threshold functions for distinct parts: revisiting erdős---lehner

Authors:
Éva Czabarka;Matteo Marsili;László A. Székely
Affiliations:
Department of Mathematics, University of South Carolina, Columbia, SC;The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy;Department of Mathematics, University of South Carolina, Columbia, SC
Venue:
Information Theory, Combinatorics, and Search Theory
Year:
2013

Citing 3
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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 Erdős---Lehner asymptotic formula for the number of partitions of the integer n into k parts with k=o(n1/3). The problem is motivated by the statistics of an experiment, where we only can tell whether outcomes are identical or different.