Abstract
We construct partition ideals with highly oscillating growth function. Partition ideal X is a set of integer partitions closed under the subpartition relation. Its growth function p(n,X) counts the number of partitions of n that are in X. We present a partition ideal with growth function being infinitely many times both zero and close to the partition function p(n) of all partitions of n.Our main result is based on the strong form of the Hardy–Ramanujan asymptotics for p(n), from which we deduce the asymptotics of p−S(n), the number of partitions of n that do not use parts from a finite set S of positive integers.
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