Partition analysis and the little Göllnitz identites

TL;DR

The paper addresses refining and unifying partition identities tied to the little G"ollnitz identities and Andrews' mod $8$ partition identity through MacMahon's Partition Analysis. It develops refined generating functions that track part sizes via the Omega operator, producing explicit series and product forms for multiple partition families: $\mathcal{G}_1,\mathcal{G}_2,\mathcal{G}'_1,\mathcal{G}'_2,\mathcal{P}_1,\mathcal{P}_2,\mathcal{P}'_1,\mathcal{P}'_2$, and demonstrates their applications to partition statistics such as the alternating sum $a(\lambda)$ and Schmidt weight $S(\lambda)$. The work also connects these refinements to classical $q$-series identities like $q$-Gauss and $q$-Lebesgue, showing how specialization recovers known sum- and product-sides and enabling new probabilistic or statistical interpretations. Overall, it provides a flexible framework for refined combinatorial interpretations of modular partition identities and opens avenues for generalization and deeper statistic-driven analyses.

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 Göllnitz 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.

Theorems & Definitions (46)

• Theorem 1.1: Göllnitz
• Theorem 1.2: Savage and Sills
• Theorem 1.3: Andrews' mod $8$ identity
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Remark 1.8
• Definition 2.1
• Lemma 2.2