  Recently, Rattan and the first author (Ann. Comb. 25 (2021) 697-728) proved a
conjectured inequality of Berkovich and Uncu (Ann. Comb. 23 (2019) 263-284)
concerning partitions with an impermissible part. In this article, we
generalize this inequality upon considering t impermissible parts. We compare
these with partitions whose certain parts appear with a frequency which is a
perfect t^{th} power. Our inequalities hold after a certain bound, which for
given t is a polynomial in s, a major improvement over the previously known
bound in the case t=1. To prove these inequalities, our methods involve
constructing injective maps between the relevant sets of partitions. The
construction of these maps crucially involves concepts from analysis and
calculus, such as explicit maps used to prove countability of N^t, and Jensen's
inequality for convex functions, and then merge them with techniques from
number theory such as Frobenius numbers, congruence classes, binary numbers and
quadratic residues. We also show a connection of our results to colored
partitions. Finally, we pose an open problem which seems to be related to power
residues and the almost universality of diagonal ternary quadratic forms.
