Asymptotics of Reciprocal Supernorm Partition Statistics

TL;DR

This paper studies the asymptotic behavior of reciprocal supernorm statistics on integer partitions, revealing they grow like e^{γ} times the logarithm of the partition size across different ensembles.

Contribution

It introduces new reciprocal supernorm statistics and establishes their asymptotic growth as e^{γ} log n for various partition ensembles.

Findings

Reciprocal supernorm statistics grow asymptotically as e^{γ} log n.

The results apply uniformly across three different partition ensembles.

The study connects partition statistics with classical constants like Euler's gamma.

Abstract

We consider two multiplicative statistics on the set of integer partitions: the norm of a partition, which is the product of its parts, and the supernorm of a partition, which is the product of the prime numbers $p_i$ indexed by its parts $i$. We introduce and study new statistics that are sums of reciprocals of supernorms on three statistical ensembles of partitions, labelled by their size $|\lambda|=n$, their perimeter equaling $n$, and their largest part equaling $n$. We show that the cumulative statistics of the reciprocal supernorm for each of the three ensembles are asymptotic to $e^{\gamma} \log n$ as $n \to \infty$.