Weighted cylindric partitions

TL;DR

This paper extends a framework for deriving sum-product identities to include weighted and symmetric cylindric partitions, leading to new identities and proofs of classical results in partition theory.

Contribution

It introduces a generalized approach to generating identities involving weighted cylindric partitions and related structures, expanding the scope of previous methods.

Findings

Proved new identities involving weighted cylindric partitions.

Provided new proofs for classical identities like G"ollnitz-Gordon.

Extended the framework to include symmetric and skew double shifted plane partitions.

Abstract

Recently Corteel and Welsh outlined a technique for finding new sum-product identities by using functional relations between generating functions for cylindric partitions and a theorem of Borodin. Here, we extend this framework to include very general product-sides coming from work of Han and Xiong. In doing so, we are led to consider structures such as weighted cylindric partitions, symmetric cylindric partitions and weighted skew double shifted plane partitions. We prove some new identities and obtain new proofs of known identities, including the G\"ollnitz-Gordon and Little G\"ollnitz identities as well as some beautiful Schmidt-type identities of Andrews and Paule.