The number of equivalence classes arising from partition involutions

Involutions have played important roles in many research areas including the theory of partitions. In this paper, for various sets of partitions, we give relations between the number of equivalence classes in the set of partitions arising from an involution and the number of partitions satisfying a certain parity condition. We examine the number of equivalence classes arising from the conjugations on ordinary partitions, overpartitions, and partitions with distinct odd parts. We also consider other types of involutions on partitions into distinct parts, unimodal sequences with a unique marked peak, and partitions with distinct even parts.