Some Parity Statistics in Integer Partitions

We study integer partitions with respect to the classical word statistics of levels and descents subject to prescribed parity conditions. For instance, a partition with summands $\lambda _1\ge \cdots \ge \lambda _k$ may be enumerated according to descents $\lambda _i>\lambda _{i+1}$ while tracking the individual parities of $\lambda _i$ and $\lambda _{i+1}$. There are two types of parity levels, $E=E$ and $O=O$, and four types of parity-descents, $E>E$, $E>O$, $O>E$ and $O>O$, where $E$ and $O$ represent arbitrary even and odd summands. We obtain functional equations and explicit generating functions for the number of partitions of $n$ according to the joint occurrence of the two levels. Then we obtain corresponding results for the joint occurrence of the four types of parity-descents. We also provide enumeration results for the total number of occurrences of each statistic in all partitions of $n$ together with asymptotic estimates for the average number of parity-levels in a random partition.