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Summary
This summary is machine-generated.

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.

Keywords:
Cylindric partitionsPartition diamondsPartition identitiesSkew double-shifted plane partitionsWeighted partition identities

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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.