Mircea Merca
University Politehnica of Bucharest, Bucharest, Romania

The first appearance of parity in partitions arose in Legendre's interpretation of Euler's pentagonal number theorem. Since then, the parity of parts has played a central role in many works on partitions. We investigate the number of the parts with the same parity in all the partitions of $n$ and provide connections with other counting functions. As applications of some truncated theta series, we introduce a collection of identities and infinite families of linear inequalities involving the number of the parts with the same parity in all the partitions of $n$. Moreover, we provide connections with partitions with non-negative rank, partition with non-negative crank and Garden of Eden partitions. Additive evaluations of some multiplicative functions are investigated in this context.