Generalized Alder-Type Partition Inequalities

TL;DR

This paper proves a conjecture related to Alder-type partition inequalities for large parameters, extending previous results and generalizing the shift inequality to arbitrary shifts, advancing the understanding of partition identities.

Contribution

The paper proves a conjecture for large $n$, generalizes the shift inequality for arbitrary shifts, and discusses implications for Alder-type partition inequalities.

Findings

Proved the conjecture for large enough $n$.

Generalized the shift inequality for arbitrary shifts.

Discussed implications for Alder-type inequalities.

Abstract

In 2020, Kang and Park conjectured a "level $2$" Alder-type partition inequality which encompasses the second Rogers-Ramanujan Identity. Duncan, Khunger, the fourth author, and Tamura proved Kang and Park's conjecture for all but finitely many cases utilizing a "shift" inequality and conjectured a further, weaker generalization that would extend both Alder's (now proven) as well as Kang and Park's conjecture to general level. Utilizing a modified shift inequality, Inagaki and Tamura have recently proven that the Kang and Park conjecture holds for level $3$ in all but finitely many cases. They further conjectured a stronger shift inequality which would imply a general level result for all but finitely many cases. Here, we prove their conjecture for large enough $n$, generalize the result for an arbitrary shift, and discuss the implications for Alder-type partition inequalities.