  PED partitions are partitions with even parts distinct while odd parts are
unrestricted. Similarly, POD partitions have distinct odd parts while even
parts are unrestricted. Merca proved several recurrence relations analytically
for the number of PED partitions of $n$. They are similar to the recurrence
relation for the number of partitions of $n$ given by Euler's pentagonal number
theorem. We provide combinatorial proofs for all of these theorems and also for
the pentagonal number theorem for PED partitions proved analytically by Fink,
Guy, and Krusemeyer. Moreover, we prove combinatorially a recurrence for POD
partitions given by Ballantine and Merca, Beck-type identities involving PED
and POD partitions, and several other results about PED and POD partitions.
