DMTCS Proceedings, 23rd Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA'12)

Font Size:  Small  Medium  Large

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

Dimbinaina Ralaivaosaona


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≥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.

Full Text: PostScript PDF

Valid XHTML 1.0 Transitional