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)/2m-1 for m = o(n1/5) [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)/2m-1 \leq \rho(n,m) \leq q(n+m(m-1)/2)/2m-1 and (ii)  \rho(n,m) \leq q(n+[m(m-1)/4])/2m-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])/2m-1.

Theorem 3: For fixed \epsilon, with 0 < \epsilon < 10-2 and for m = m(n), 1 \leq m \leq n3/8-\epsilon, and n ––> oo,

\rho(n,m) = (1+o(1))q(n)/prod1 \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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