## Partitions into parts which are unequal and large.(English)Zbl 0679.10013

Number theory, Proc. 15th Journ. Arith., Ulm/FRG 1987, Lect. Notes Math. 1380, 19-30 (1989).
[For the entire collection see Zbl 0667.00007.]
Let $$q(n)$$ be the number of partitions of $$n$$ into unequal parts, and let $$\rho(n,m)$$ be the number of partitions of $$n$$ into unequal parts $$\geq m$$. The first and third authors have previously shown that $$\rho(n,m)=(1+o(1))q(n)/2^{m-1}$$ for $$m=o(n^{1/5})$$ [Colloq. Math. Soc. János Bolyai 34, 397-450 (1984; Zbl 0548.10010)]. Three additional theorems giving estimates for $$\rho(n,m)$$ are now obtained.
Theorem 1: For all $$n\geq 1$$ and $$m$$ such that $$1\leq m\leq n$$, we have $$(i)\quad q(n)/2^{m-1}\leq \rho(n,m)\leq q(n+m(m-1)/2)/2^{m-1}$$ and $$(ii)\quad \rho(n,m)\leq q(n+[m(m-1)/4])/2^{m-2},$$ where $$[x]$$ is the integral part of $$x$$.
Theorem 2: When n tends to infinity, and $$m=o(n/\log n)^{1/3}$$, we have $\rho (n,m)=(1+o(1))q(n+[m(m-1)/4])/2^{m-1}.$ Theorem 3: For fixed $$\epsilon$$, with $$0<\epsilon <10^{-2}$$ and for $$m=m(n)$$, $$1\leq m\leq n^{3/8-\epsilon}$$, and $$n\to \infty$$, $\rho(n,m)=(1+o(1))q(n)/\prod_{1\leq j\leq m-1}(1+\exp(-\pi j/2\sqrt{3n})).$ The paper concludes with a table of values for $$\rho(n,m)$$ with $$1\leq n\leq 100$$ and $$1\leq m\leq \min (n,12)$$.
Reviewer: B.Garrison

### MSC:

 11P81 Elementary theory of partitions

### Keywords:

partitions with unequal parts; number of partitions; table

### Citations:

Zbl 0667.00007; Zbl 0548.10010