Gentile statistics and restricted partitions

TL;DR

This paper generalizes asymptotic formulas for integer partitions, connecting number theory with quantum states, to include restricted partitions where each summand can occur at most k times, extending previous results.

Contribution

It introduces an asymptotic formula for restricted partitions p_k^s(n), generalizing prior results to arbitrary powers s and maximum occurrences k.

Findings

Reproduces known asymptotics for s=1

Provides new asymptotic results for arbitrary s

Connects partition theory with quantum density of states

Abstract

In a recent paper (Tran et al., Ann.Phys.311(2004)204), some asymptotic number theoretical results on the partitioning of an integer were derived exploiting its connection to the quantum density of states of a many-particle system. We generalise these results to obtain an asymptotic formula for the restricted or coloured partitions p_k^s(n), which is the number of partitions of an integer n into the summand of s^{th} powers of integers such that each power of a given integer may occur utmost k times. While the method is not rigorous, it reproduces the well known asymptotic results for s=1 apart from yielding more general results for arbitrary values of s.