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 λ1≥⋯≥λk may be enumerated according to descents λi>λi+1 while tracking the individual parities of λi and λi+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.