Partitions with Parts Separated by Parity

There have been a number of papers on partitions in which the parity of parts plays a central role. In this paper, the parts of partitions are separated by parity, either all odd parts are smaller than all even parts or vice versa. This concept first arose in a study related to the third-order mock theta function (q). The current study also leads back to one of the Ramanujan's more mysterious functions.