Overview
This study investigates skew double-shifted plane partitions (DSPPs) which possess three-element profiles. The primary analytical tool employed is MacMahon's partition analysis. Generating function formulas are developed for these partitions, incorporating an additional constraint related to the number of non-zero diagonal elements. The research also explores infinite hierarchies of $q$-series identities derived from these formulas and examines palindromic properties within the generating functions. Furthermore, the objects are analyzed through the lens of linear partitions, which yields additional $q$-series identities. The study notes a relationship between these objects and the Göllnitz-Gordon and little Göllnitz identities.
Approach
The research methodology is centered on the application of MacMahon's partition analysis to understand skew double-shifted plane partitions. This analytical technique is utilized to derive new generating function formulas. A specific bound on the number of non-zero diagonal elements is integrated into these formulas. The approach further involves the investigation of infinite hierarchies of $q$-series identities that emerge as a consequence of the developed formulas. An examination of palindromic properties within these generating functions is also conducted. Finally, the study re-evaluates the same objects from the perspective of linear partitions, a distinct approach leading to new $q$-series identities.
Findings
- New generating function formulas have been presented for skew double-shifted plane partitions with three-element profiles.
- These formulas incorporate an additional bound specifically for the number of non-zero diagonal elements.
- The studied objects demonstrate a close relationship to the Göllnitz-Gordon and little Göllnitz identities.
- Infinite hierarchies of $q$-series identities stem from the derived generating function formulas.
- Palindromic properties have been observed within these generating functions.
- Examination of these objects from the perspective of linear partitions leads to new $q$-series identities.