On the distribution of the norm of partitions

TL;DR

This paper investigates the distribution of the norm of integer partitions, defined as the product of parts, using moment methods and singularity analysis, revealing it lacks a non-trivial limiting distribution as partition size grows.

Contribution

It introduces a novel analysis of the partition norm distribution, applying advanced asymptotic techniques to establish its limiting behavior.

Findings

The norm's moments grow asymptotically as shown by singularity analysis.

The norm does not have a non-trivial limiting distribution on [0,∞).

The study connects partition norms to partition zeta functions.

Abstract

The norm of an integer partition is defined as the product of its parts. This statistic was recently introduced by Schneider in connection to partition zeta functions. In this note, we use the method of moments to study the distribution of the norm under the uniform probability measure on partitions of $n$ as $n \to \infty$. We use singularity analysis to prove asymptotics for the moments and show as a result that the norm lacks a non-trivial limiting distribution on $[0,\infty)$.