A MacMahon analysis view of cylindric partitions

Area of Science:

• Combinatorics
• Number Theory
• q-series

Background:

• Cylindric partitions are a generalization of plane partitions.
• MacMahon's partition analysis provides a powerful framework for studying partition problems.

Purpose of the Study:

• To derive explicit formulas for generating functions of cylindric partitions.
• To explore related q-series identities and generalize known results.
• To discover new polynomial identities in the field of partition theory.

Main Methods:

• Utilizing MacMahon's partition analysis to find recurrences for cylindric partitions.
• Solving these recurrences to obtain generating functions.
• Applying a variant of the Bailey lemma to establish new identities.

Main Results:

• Explicit formulas for generating functions of cylindric partitions.
• Demonstration of the positive nature of certain alternating q-series.
• Generalization of existing partition identities (Andrews-Gordon, Bressoud).
• Discovery of numerous new infinite hierarchies of polynomial identities.

Conclusions:

• The study provides a systematic method for analyzing cylindric partitions.
• New q-series and polynomial identities are established, expanding the landscape of partition theory.
• The findings offer insights into the structure and properties of combinatorial objects.