## Partitions into distinct large parts.(English)Zbl 0824.11064

Let $$q_ m (n)$$ denote the number of partitions of the positive integer $$n$$ into distinct parts, each of which is at least $$m$$. As to the relatively small values of $$m$$, P. Erdős and the reviewer [Topics in classical number theory, Colloq. Budapest 1981, Colloq. Math. Soc. J. Bolyai 34, 397-450 (1984; Zbl 0548.10010)] observed that $$q_ m (n)\sim q_ 1 (n)/ 2^{m-1}$$ for $$1\leq m\leq n^{1/5}$$ and $$n\to\infty$$. P. Erdős, J.-L. Nicolas and the reviewer [Number theory (Ulm, 1987), Lect. Notes Math. 1380, 19-30 (1989; Zbl 0679.10013)] proved that $$q_ m (n)\sim q_ 1 (n) \prod_{j=1}^{m-1} (1+ \exp (-\pi j(12n)^{- 1/2}) )^{-1}$$ for $$m\leq n^{3/8- \varepsilon}$$, $$\varepsilon>0$$.
In the paper under review the authors give an asymptotic estimate of $$q_ m (n)$$ as $$n\to \infty$$, valid for $$m= o(n \log^{-9} n)$$. However, the main term of the estimate for $$q_ m (n)$$ in Theorem 1 involves a parameter which is not given explicitly in terms of $$m$$ and $$n$$. Theorem 2 yields an explicit asymptotic estimate for $$q_ m (n)$$ which is valid for $$m$$ relatively large compared with $$n^{1/2}$$ and involves the inverse of the function $$x^{-2} \int_ x^ \infty y(1+ \exp (y))^{-1} dy$$.
As a corollary, the authors obtain a family of asymptotic estimates for $$q_ m (n)$$ in terms of elementary functions, each estimate being valid for large $$m$$ in a specified interval of length at least $$n^{1/3}$$. Finally, Theorem 3 refers to $$m=o (n^{1/3})$$.

### MSC:

 11P82 Analytic theory of partitions

### Keywords:

partitions into parts which are unequal and large

### Citations:

Zbl 0548.10010; Zbl 0679.10013