Polynomial partition asymptotics

Let f∈Z[y] be a polynomial such that f(N)⊆N, and let pAf(n) denote number of partitions of n whose parts lie in the set Af:={f(n):n∈N}. Under hypotheses on the roots of f−f(0), we use the Hardy-Littlewood circle method, a polylogarithm identity, and the Matsumoto-Weng zeta function to derive asymptotic formulae for pAf(n) as n tends to infinity. This generalises asymptotic formulae for the number of partitions into perfect dth powers, established by Vaughan for d=2, and Gafni for the case d≥2, in 2015 and 2016 respectively.