Concerning periodicity in the asymptotic behaviour of partition functions

Abstract

Let P_A(n) denote the number of partitions of n into summands chosen from the set A = {a₁, a₂, …}. De Bruijn has shown that in Mahler's partition problem (a_ν = r^ν) there is a periodic component in the asymptotic behaviour of P_A(n). We show by example that this may happen for sequences that satisfy a_ν ν and consider an analogous phenomena for partitions into primes. We then consider corresponding results for partitions into distinct summands. Finally we obtain some weaker results using elementary methods.