  Numerous congruences for partitions with designated summands have been proven
since first being introduced and studied by Andrews, Lewis, and Lovejoy. This
paper explicitly characterizes the number of partitions with designated
summands whose parts are not divisible by $2^\ell$, $2$, and $3^\ell$ working
modulo $2,\ 4,$ and $3$, respectively, greatly extending previous results on
the subject. We provide a few applications of our characterizations throughout
in the form of congruences and a computationally fast recurrence. Moreover, we
illustrate a previously undocumented connection between the number of
partitions with designated summands and the number of partitions with odd
multiplicities.
