Unimodality of Partitions in Near-Rectangular Ferrers Diagrams

TL;DR

This paper investigates the unimodality of the rank generating function of partitions within Ferrers diagrams, extending known results to larger classes of partitions and identifying conditions for unimodality.

Contribution

It extends the class of 4-part partitions for which the unimodality of the generating function is known and establishes new conditions ensuring unimodality.

Findings

G_lambda is not unimodal for larger classes of 4-part partitions.

G_lambda is unimodal when parts are close in size or the first part is at least half of the total.

The results generalize previous findings on unimodality in Ferrers diagram partitions.

Abstract

We look at the rank generating function $G_\lambda$ of partitions inside the Ferrers diagram of some partition $\lambda$, investigated by Stanton in 1990, as well as a closely related problem investigated by Stanley and Zanello in 2013. We show that $G_\lambda$ is not unimodal for a larger class of 4-part partitions than previously known, and also that if the ratios of parts of $\lambda$ are close enough to 1 (depending on how many parts $\lambda$ has), or if the first part is at least half the size of $\lambda$, then $G_\lambda$ is unimodal.