Partitions into prime powers

For a subset \(\mathcal A\subset \mathbb N\), let \(p_{\mathcal A}(n)\) denote the restricted partition function which counts partitions of \(n\) with all parts lying in \(\mathcal A\). In this paper, we use a variation of the Hardy-Littlewood circle method to provide an asymptotic formula for \(p_{\mathcal A}(n)\), where \(\mathcal A\) is the set of \(k\)-th powers of primes (for fixed \(k\)). This combines Vaughan's work on partitions into primes with the author's previous result about partitions into \(k\)-th powers. This new asymptotic formula is an extension of a pattern indicated by several results about restricted partition functions over the past few years. Comparing these results side-by-side, we discuss a general strategy by which one could analyze \(p_{\mathcal A}(n )\) for a given set \(\mathcal A\).