Berkovich-Uncu type Partition Inequalities Concerning Impermissible Sets and Perfect Power Frequencies

TL;DR

This paper generalizes partition inequalities involving impermissible parts and perfect power frequencies, using advanced combinatorial, analytical, and number-theoretic techniques, and improves bounds significantly for multiple impermissible parts.

Contribution

It extends previous inequalities to multiple impermissible parts, introduces polynomial bounds, and employs novel injective map constructions combining analysis and number theory.

Findings

Inequalities hold after polynomial bounds in s for given t

Constructed injective maps using analysis and number theory

Connected results to colored partitions and posed open problems

Abstract

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.