A phase transition in the distribution of the length of integer partitions

We assign a uniform probability to the set consisting of partitions of a positive integer $n$ such that the multiplicity of each summand is less than a given number $d$ and we study the limiting distribution of the number of summands in a random partition. It is known from a result by Erdős and Lehn...

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Bibliographic Details
Published in:Discrete mathematics and theoretical computer science Vol. DMTCS Proceedings vol. AQ,...; no. Proceedings; pp. 265 - 282
Main Author: Ralaivaosaona, Dimbinaina
Format: Journal Article Conference Proceeding
Language:English
Published: DMTCS 01.01.2012
Discrete Mathematics and Theoretical Computer Science
Discrete Mathematics & Theoretical Computer Science
Series:DMTCS Proceedings
Subjects:
[info.info-cg] computer science [cs]/computational geometry [cs.cg]
[info.info-dm] computer science [cs]/discrete mathematics [cs.dm]
[info.info-ds] computer science [cs]/data structures and algorithms [cs.ds]
[math.math-co] mathematics [math]/combinatorics [math.co]
asymptotic expansions
Combinatorics
Computational Geometry
Computer Science
Data Structures and Algorithms
Discrete Mathematics
integer partitions
limit distribution
Mathematics
multiplicities
limit distribution
Asymptotic expansions
multiplicities
integer partitions
ISSN:1365-8050, 1462-7264, 1365-8050
Online Access:Get full text
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Summary:We assign a uniform probability to the set consisting of partitions of a positive integer $n$ such that the multiplicity of each summand is less than a given number $d$ and we study the limiting distribution of the number of summands in a random partition. It is known from a result by Erdős and Lehner published in 1941 that the distributions of the length in random restricted $(d=2)$ and random unrestricted $(d \geq n+1)$ partitions behave very differently. In this paper we show that as the bound $d$ increases we observe a phase transition in which the distribution goes from the Gaussian distribution of the restricted case to the Gumbel distribution of the unrestricted case.
ISSN:1365-8050
1462-7264
1365-8050
DOI:10.46298/dmtcs.2999