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Weighted cylindric partitions
Walter Bridges1, Ali K Uncu2,3
1Department of Mathematics and Computer Science, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany.
Journal of Algebraic Combinatorics
|October 19, 2022
Summary
This study expands on finding sum-product identities using generating functions for cylindric partitions. New identities and proofs for known ones, like Göllnitz-Gordon, are achieved by incorporating general product-sides.
Area of Science:
- Combinatorics
- Number Theory
- Algebraic Combinatorics
Background:
- Cylindric partitions are combinatorial objects studied for their generating functions.
- Previous work by Corteel and Welsh established a method for discovering sum-product identities using these functions.
- General product-sides from Han and Xiong's work offer potential for new identity discoveries.
Purpose of the Study:
- To extend the framework for finding sum-product identities by incorporating general product-sides.
- To explore new combinatorial structures like weighted and symmetric cylindric partitions.
- To derive new identities and provide novel proofs for existing ones.
Main Methods:
- Utilizing functional relations between generating functions for cylindric partitions.
- Extending Corteel and Welsh's technique to include general product-sides.
- Analyzing structures such as weighted cylindric partitions, symmetric cylindric partitions, and weighted skew double-shifted plane partitions.
Main Results:
- New sum-product identities have been proven.
- New proofs for known identities, including the Göllnitz-Gordon and Little Göllnitz identities, have been obtained.
- Several Schmidt-type identities by Andrews and Paule are derived.
Conclusions:
- The extended framework effectively generates new sum-product identities.
- The study contributes to the understanding of cylindric partitions and related combinatorial structures.
- This work offers elegant new proofs for significant identities in the field.
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