Verification and generation of unrefinable partitions

Unrefinable partitions are a subset of partitions into distinct parts which satisfy an additional unrefinability property. More precisely, no parts of such partitions can be written as the sum of different integers which are not parts. We address in this paper the algorithmic aspects related to unrefinable partitions, such as testing whether a given partition is unrefinable or not and enumerating all the partitions whose sum is a given number. We design two algorithms to solve the two mentioned problems and we discuss their complexity.