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Published in:
2013 | OriginalPaper | Chapter
Threshold Functions for Distinct Parts: Revisiting Erdős–Lehner
Authors : Éva Czabarka, Matteo Marsili, László A. Székely
Published in: Information Theory, Combinatorics, and Search Theory
Publisher: Springer Berlin Heidelberg
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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
(
n
1/3
). The problem is motivated by the statistics of an experiment, where we only can tell whether outcomes are identical or different.