## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  673.05007
Autor:  Erdös, Paul; Nicolas, J.L.; Sárközy, A.
Title:  On the number of partitions of n without a given subsum. I. (In English)
Source:  Discrete Math. 75, No.1-3, 155-166 (1989).
Review:  Let R(n,a) be the number of partitions of n = n1+...+nt, where subsums ni1+...+nij are all different from a. The authors examine R(n,a) for a depending on n and smaller than \lambda0\sqrt{n}, \lambda0 a small positive constant. They prove that there exists \lambda0 > 0 such that uniformly for 1 \leq a \leq \lambda0\sqrt{n}, when n goes to infinity,

(1)  log (\frac{R(n,a)}{p(n)}) \leq (\psi(a) log \frac{\pi a}{\sqrt{6n}})+O(1/\sqrt{n}),

(2)  log\binom{R(n,a)}{p(n)} \geq \psi(a) log \frac{\pi a}{\sqrt{6n}}-\gammaaa+O(a2/\sqrt{n})

where \psi(a) = \lfloor a/2 +1\rfloor, p(n) is the unrestricted partition function and \gammaa = ½ if a is odd, and \gammaa = ½+ log 3-(7/6) log 2+(c log a)/a = +0.79...+(c log a)/a if a is even, where c is a fixed constant.
Further, if Q(n,a) is the above function R(n,a) with the restriction that each part occurs at most once, then there exists \lambda1 > 0 such that uniformly for 1 \leq a \leq \lambda1\sqrt{n}, log(\frac{Q(n,a)}{q(n)}) \geq -(a/6) log (16/3)- log 3+O(a2/\sqrt{n}).
Reviewer:  G.L.Alexanderson
Classif.:  * 05A17 Partitions of integres (combinatorics)
05A15 Combinatorial enumeration problems
11P81 Elementary theory of partitions
Keywords:  number of partitions

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