Unimodality of ranks and a proof of Stanton's conjecture

TL;DR

This paper proves the unimodality of ranks in partition functions using the Circle Method and confirms Stanton's conjecture, advancing understanding of the positivity and structure of related polynomials.

Contribution

It introduces a proof of Stanton's conjecture on polynomial positivity and demonstrates the unimodality of ranks in partition functions.

Findings

Proved unimodality of ranks using the Circle Method.

Confirmed Stanton's conjecture on polynomial positivity.

Established groundwork for future combinatorial structure analysis.

Abstract

Recently, much attention has been given to various inequalities among partition functions. For example, Nicolas, {and later DeSavlvo--Pak,} proved that $p(n)$ is eventually log-concave, and Ji--Zang showed that the cranks are eventually unimodal. This has led to a flurry of recent activity generalizing such results in different directions. At the same time, Stanton recently made deep conjectures on the positivity of certain polynomials associated to ranks and cranks of partitions, with the ultimate goal of pointing the way to ``deeper'' structure refining ranks and cranks. These have been shown to be robust in recent works, which have identified further infinite families of such conjectures in the case of colored partitions. In this paper, we employ the Circle Method to prove unimodality for ranks. As a corollary, we prove Stanton's original conjecture. This points to future study of the positive, integral coefficients Stanton conjectured to exist, hinting at new combinatorial structure yet to be uncovered.