On the asymptotic behaviour of general partition functions. II (Q1417938)

Let \(A\subset \mathbb{N}\), where \(\mathbb{N}\) denotes the set of all natural numbers. If \(n\in \mathbb{N}\), let \(p_A(n)\), \(q_A(n)\) denote respectively the number of partitions of \(n\) into parts, distinct parts from \(A\). Among other results, the authors prove the existence of an infinite subset \(S\subset \mathbb{N}\) such that \[ \liminf_{n\to\infty}\, {\log p_S(n)\over\log q_S(n)}= 1. \] For Part I, cf. ibid. 4, 29--39 (2000; Zbl 1010.11059).