Analogues of Alder-Type Partition Inequalities for Fixed Perimeter Partitions

TL;DR

This paper extends Alder-type partition inequalities to fixed perimeter partitions, providing new generalized inequalities and a reverse inequality, building on prior work related to Euler's and Rogers-Ramanujan identities.

Contribution

It introduces generalized Alder-type inequalities for fixed perimeter partitions and establishes a reverse inequality, advancing the theoretical understanding of partition identities.

Findings

Established generalized Alder-type inequalities for fixed perimeter partitions.

Proved a reverse Alder-type inequality in the same setting.

Extended previous results by Fu and Tang to new partition classes.

Abstract

In a 2016 paper, Straub proved an analogue to Euler's partition identity for partitions with fixed perimeter. Later, Fu and Tang provided a refinement and generalization of Straub's analogue to $d$-distinct partitions as well as a result related to the first Rogers-Ramanujan identity. Motivated by Alder-type partition identities and their generalizations, we build on work of Fu and Tang to establish generalized Alder-type partition inequalities in a fixed perimeter setting, and notably, a reverse Alder-type inequality.