Gentile statistics and restricted partitions

摘要

In a recent paper (Tran et al, Ann. Phys. 311, 204 (2004)), 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 e?‘? into the summand of e?‘?th powers of integers such that each power of a given integer may occur utmost e?‘? times. While the method is not rigorous, it reproduces the well-known asymptotic results for e?‘? = 1 apart from yielding more general results for arbitrary values of e?‘?.