Higher Order Smallest Parts Functions and Rank-Crank Moment Inequalities from Bailey Pairs

TL;DR

This paper generalizes inequalities related to partition rank and crank moments using Bailey pairs, introduces higher order smallest parts functions, and explores their properties, inequalities, and conjectures in partition theory.

Contribution

It introduces a new rank-like function and establishes inequalities for moments, extending Garvan's results and providing a framework for higher order smallest parts functions.

Findings

Established inequalities among rank-like and crank-like functions

Introduced higher order smallest parts functions with new properties

Conjectured additional inequalities and congruences for these functions

Abstract

We generalize a result of Garvan on inequalities and interpretations of the moments of the partition rank and crank functions. In particular for nearly 30 Bailey pairs, we introduce a rank-like function, establish inequalities with the moments of the rank-like function and an associated crank-like function, and give an associated so called higher order smallest parts function. In some cases we are able to deduce inequalities among the rank-like functions. We also conjecture additional inequalities and a large number of congruences for the higher order smallest parts functions.