Infinite families of crank functions, Stanton-type conjectures, and unimodality

TL;DR

This paper proves Stanton's conjectures related to crank functions, extends these conjectures to infinite families, and explores their implications for combinatorial structures and unimodality in partition theory.

Contribution

It proves two Stanton conjectures, extends them to infinite families of crank functions, and provides criteria for polynomial quotients to have non-negative coefficients.

Findings

Proof of two Stanton conjectures

Extension to infinite families of crank functions

Criteria for non-negative polynomial quotients

Abstract

Dyson's rank function and the Andrews--Garvan crank function famously give combinatorial witnesses for Ramanujan's partition function congruences modulo 5, 7, and 11. While these functions can be used to show that the corresponding sets of partitions split into 5, 7, or 11 equally sized sets, one may ask how to make the resulting bijections between partitions organized by rank or crank combinatorially explicit. Stanton recently made conjectures which aim to uncover a deeper combinatorial structure along these lines, where it turns out that minor modifications of the rank and crank are required. Here, we prove two of these conjectures. We also provide abstract criteria for quotients of polynomials by certain cyclotomic polynomials to have non-negative coefficients based on unimodality and symmetry. Furthermore, we extend Stanton's conjecture to an infinite family of cranks. This suggests further applications to other combinatorial objects. We also discuss numerical evidence for our conjectures, connections with other analytic conjectures such as the distribution of partition ranks.