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Numerous congruences for partitions with designated summands have been provensince first being introduced and studied by Andrews, Lewis, and Lovejoy. Thispaper explicitly characterizes the number of partitions with designatedsummands whose parts are not divisible by $2^\ell$, $2$, and $3^\ell$ workingmodulo $2,\ 4,$ and $3$, respectively, greatly extending previous results onthe subject. We provide a few applications of our characterizations throughoutin the form of congruences and a computationally fast recurrence. Moreover, weillustrate a previously undocumented connection between the number ofpartitions with designated summands and the number of partitions with oddmultiplicities.

Partitions With Designated Summands Not Divisible by $2^l$, $2$, and $3^l$ Modulo $2$, $4$, and $3$