A MacMahon Analysis View of Cylindric Partitions

TL;DR

This paper applies MacMahon's partition analysis to cylindric partitions with two-element profiles, deriving explicit generating functions, exploring q-series identities, and proposing new polynomial refinements and hierarchies of identities.

Contribution

It introduces explicit formulas for cylindric partition generating functions and generalizes known identities using MacMahon analysis and Bailey lemma techniques.

Findings

Derived explicit generating functions for cylindric partitions.

Identified and generalized q-series identities related to these partitions.

Proposed new polynomial refinements of classical identities.

Abstract

We study cylindric partitions with two-element profiles using MacMahon's partition analysis. We find explicit formulas for the generating functions of the number of cylindric partitions by first finding the recurrences using partition analysis and then solving them. We also note some q-series identities related to these objects that show the manifestly positive nature of some alternating series. We generalize the proven identities and conjecture new polynomial refinements of Andrews-Gordon and Bressoud identities, which are companions to Foda-Quano's refinements. Finally, using a variant of the Bailey lemma, we present many new infinite hierarchies of polynomial identities.