Partition analysis and the little G\"ollnitz identites

TL;DR

This paper uses MacMahon's Partition Analysis to refine generating functions for partitions related to the little G"ollnitz identities and explores their applications in partition statistics, extending previous work by Savage, Sills, and Andrews.

Contribution

It introduces refined generating functions for partitions connected to the little G"ollnitz identities using MacMahon's Partition Analysis, with applications to partition statistics.

Findings

Refined generating functions for partitions related to the identities.

Application of these functions to partition statistics like alternating sum and Schmidt weight.

Extension of previous identities with new partition interpretations.

Abstract

This work follows the spirit of Andrews' series of papers on Partition Analysis. In $2011$, Savage and Sills found new sum sides for the little G\"ollnitz identities and provided their partition interpretations. It turns out that similar companions exist for a mod $8$ partition identity due to Andrews. In this work, we use MacMahon's Partition Analysis to study partitions related to these identities. We find refined generating functions for them, where we keep track of the size of each part. Finally, by considering the alternating sum and Schmidt weight, we show the application of these refined functions in the study of partition statistics.