Gentile statistics and restricted partitions

摘要

In a recent paper (Tran et al., Ann.Phys.311(2004)204), some asymptoticnumber theoretical results on the partitioning of an integer were derivedexploiting its connection to the quantum density of states of a many-particlesystem. We generalise these results to obtain an asymptotic formula for therestricted or coloured partitions p_k^s(n), which is the number of partitionsof an integer n into the summand of s^{th} powers of integers such that eachpower of a given integer may occur utmost k times. While the method is notrigorous, it reproduces the well known asymptotic results for s=1 apart fromyielding more general results for arbitrary values of s.