https://hal.inria.fr/hal-01197255Ralaivaosaona, DimbinainaDimbinainaRalaivaosaonaDMS - Department of Mathematical Sciences [Matieland, Stellenbosch Uni.] - Stellenbosch UniversityA phase transition in the distribution of the length of integer partitionsHAL CCSD2012Asymptotic expansionsinteger partitionsmultiplicitieslimit distribution.[INFO.INFO-DS] Computer Science [cs]/Data Structures and Algorithms [cs.DS][INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM][MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO][INFO.INFO-CG] Computer Science [cs]/Computational Geometry [cs.CG]Episciences Iam, CoordinationBroutin, Nicolas and Devroye, Luc2015-09-11 13:22:142017-03-07 15:19:082015-09-11 13:36:39enConference papershttps://hal.inria.fr/hal-01197255/document10.46298/dmtcs.2999application/pdf1We 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 ErdÅ‘s 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.