Asymptotics of the number partitioning distribution

TL;DR

This paper analyzes the asymptotic behavior of the number partitioning distribution by leveraging an analogy with a Bose gas, revealing that the distribution approaches its limit only for extremely large integers exceeding 10^10.

Contribution

It introduces a novel physical analogy to study the asymptotics of number partitioning distributions, providing insights into their convergence properties.

Findings

Distribution approaches asymptotics only for n > 10^10

Uses Bose gas analogy to analyze partitioning

Characterizes probability distribution for number of summands

Abstract

The number partitioning problem can be interpreted physically in terms of a thermally isolated non-interacting Bose gas trapped in a one-dimensional harmonic oscillator potential. We exploit this analogy to characterize, by means of a detour to the Bose gas within the canonical ensemble, the probability distribution for finding a specified number of summands in a randomly chosen partition of an integer n. It is shown that this distribution approaches its asymptotics only for n > 10^10.