On the asymptotic behaviour of general partition functions (Q1567057)

If \(N\) is the set of all natural numbers, \(n\in N\), and \(A \subset N\), let \(p_A(n)\) denote the number of partitions of \(n\) such that all parts belong to \(A\); likewise let \(q_A(n)\) denote the number of partitions of \(n\) into distinct parts belonging to \(A\). The authors prove four asymptotic results, one of which is as follows: Theorem 1: For every infinite set \(A\subset N\), we have \[ \limsup_{n \to\infty} {\log\biggl( \max \bigl(2,p_A(n) \bigr)\biggr) \over\log \biggl(\max \bigl(2,q_A(n) \bigr)\biggr)} \geq\sqrt 2. \] The other results require additional assumptions on \(A\).