Berkovich–Uncu Type Partition Inequalities Concerning Impermissible Sets and Perfect Power Frequencies

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