The Distribution of the Product of Parts in Integer Partitions

Given a positive integer n, a partition L = (ℓ1, ℓ2, ℓ3, . . .) is a sequence of non-decreasing positive integers whose sum is n. The norm N(L) of a partition L is defined as the product of its parts. Consider a random partition L of n sampled uniformly from all partitions of n. It was recently proved by Bridges and Craig that N(L) lacks a non-trivial limiting distribution as n → ∞. We believe this is because the norm is, in a sense, a multiplicative statistic on partitions. Hence, in this work, we study instead the logarithm of the norm (or log-norm) log N(L). We prove that the log-norm of a random partition of n converges to a continuous limiting distribution as n → ∞. We extend this result to the case of Λ-partitions, where the parts of the partitions are restricted to elements of a fixed sequence of positive integers Λ. Under a general analytic framework due to Meinardus, we are also able to show the existence of a limiting distribution for the log-norm of a random Λ-partition. Notable examples include the cases of square partitions and prime partitions. Furthermore, we consider a non-uniform measure on the set of partitions of n, defined using a Vershik-style multiplicative weight. In this setting, we find that the limiting behavior of the distribution of the log-norm undergoes a phase transition: it changes from non-Gaussian to Gaussian. Finally, for the case of restricted partitions, where parts are not allowed to repeat, we prove that the distribution is almost always Gaussian.

Grantor dc:publisher

Stellenbosch : Stellenbosch University

Year dc:date.issued

2026

Author dc:creator

• Gikunda, Dennis Kinoti

Advisor dc:contributor.advisor

• Ralaivaosaona, Dimbinaina

Language dc:language.iso

en

Repository record dc:identifier.uri

https://scholar.sun.ac.za/handle/10019.1/136012

OAI identifier oai:identifier

oai:scholar.sun.ac.za:10019.1/136012