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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 679.10013

**Autor: ** Erdös, Paul; Nicolas, J.L.; Szalay, M.

**Title: ** Partitions into parts which are unequal and large. (In English)

**Source: ** Number theory, Proc. 15th Journ. Arith., Ulm/FRG 1987, Lect. Notes Math. 1380, 19-30 (1989).

**Review: ** [For the entire collection see Zbl 667.00007.]

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 548.10010)]. 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) q(n)/2^{m-1} \leq \rho(n,m) \leq q(n+m(m-1)/2)/2^{m-1} and (ii) \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 ––> oo,

\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 **max** (n,12).

**Reviewer: ** B.Garrison

**Classif.: ** * 11P81 Elementary theory of partitions

**Keywords: ** partitions with unequal parts; number of partitions; table

**Citations: ** Zbl 667.00007; Zbl 548.10010

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