On the number of partitions of \(n\) without a given subsum. I
Let \(R(n,a)\) be the number of partitions of \(n=n_1+...+n_t\), where subsums \(n_{i_1}+...+n_{i_j}\) are all different from \(a\). The authors examine \(R(n,a)\) for a depending on \(n\) and smaller than \(\lambda_0\sqrt{n}\), \(\lambda_0\) a small positive constant. They prove that there exists \(\lambda_0>0\) such that uniformly for \(1\leq a\leq \lambda_0\sqrt{n}\), when \(n\) goes to infinity, \[ (1)\quad \log (\frac{R(n,a)}{p(n)})\leq (\psi(a) \log \frac{\pi a}{\sqrt{6n}})+O(1/\sqrt{n}), \] \[ (2)\quad \log\binom{R(n,a)}{p(n)} \geq \psi(a) \log \frac{\pi a}{\sqrt{6n}}-\gamma_aa+O(a^2/\sqrt{n}) \] where \(\psi(a)=\lfloor \frac{a}{2}+1\rfloor\), p(n) is the unrestricted partition function and \(\gamma_a=1/2\) if a is odd, and \(\gamma_a=1/2+\log 3-(7/6)\log 2+(c \log a)/a=+0.79...+(c \log a)/a\) 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 \(\lambda_1>0\) such that uniformly for \(1\leq a\leq \lambda_1\sqrt{n},\) \(\log(\frac{Q(n,a)}{q(n)})\geq -(a/6)\log (16/3)-\log 3+O(a^2/\sqrt{n}).\)
scientific article; zbMATH DE number 4101192