Construction of Evidently Positive Series and An Alternative Construction for a Family of Partition Generating Functions due to Kanade and Russell

TL;DR

This paper introduces a new combinatorial approach to construct evidently positive series for certain partition generating functions, replacing jagged partitions with ordinary partitions and providing explicit positive series for key infinite products.

Contribution

It offers an alternative combinatorial construction for partition generating functions, simplifying the representation by using ordinary partitions and deriving new positive series.

Findings

New evidently positive series for partition functions

A combinatorial decomposition method for partitions

Explicit positive series for key infinite products

Abstract

We give an alternative construction for a family of partition generating functions due to Kanade and Russell. In our alternative construction, we use ordinary partitions instead of jagged partitions. We also present new generating functions which are evidently positive series for partitions due to Kanade and Russell. To obtain those generating functions, we first construct an evidently positive series for a key infinite product. In that construction, a series of combinatorial moves is used to decompose an arbitrary partition into a base partition together with some auxiliary partitions that bijectively record the moves.