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,
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
