Weighted cylindric partitions

Walter Bridges; Ali K. Uncu

doi:10.1007/s10801-022-01156-9

Weighted cylindric partitions

Walter Bridges, Ali K. Uncu

Research output: Contribution to journal › Article › peer-review

2 Citations (SciVal)

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 Göllnitz–Gordon and Little Göllnitz identities as well as some beautiful Schmidt-type identities of Andrews and Paule.

Original language	English
Pages (from-to)	1309-1337
Number of pages	29
Journal	Journal of Algebraic Combinatorics
Volume	56
Issue number	4
Early online date	26 Aug 2022
DOIs	https://doi.org/10.1007/s10801-022-01156-9
Publication status	Published - Dec 2022

Bibliographical note

Research of the first author is partially supported by the DFG (Project Number 281071066 TRR 191). Research of the second author is partly supported by EPSRC Grant Number EP/T015713/1 and partly by FWF Grant P-34501N.

Funding Information:
We would like to thank Krishnaswami Alladi, George E. Andrews, Jehanne Dousse, Guoniu Han, Ralf Hemmecke, and Peter Paule for their interest and encouragement. We would like to particularly thank Alexander Berkovich, Christian Krattencthaler, Ole Warnaar, and Wadim Zudilin for the discussion and their valuable feedback. We would also like to thank the anonymous referees for their careful reading and suggestions. The first author is partially supported by the SFB/TRR 191 “Symplectic Structures in Geometry, Algebra and Dynamics,” funded by the DFG (Project number 281071066 TRR 191). Research of the second author is partly supported by EPSRC Grant number EP/T015713/1 and partly by FWF Grant P-34501N.

Funding Information:
We would like to thank Krishnaswami Alladi, George E. Andrews, Jehanne Dousse, Guoniu Han, Ralf Hemmecke, and Peter Paule for their interest and encouragement. We would like to particularly thank Alexander Berkovich, Christian Krattencthaler, Ole Warnaar, and Wadim Zudilin for the discussion and their valuable feedback. We would also like to thank the anonymous referees for their careful reading and suggestions. The first author is partially supported by the SFB/TRR 191 “Symplectic Structures in Geometry, Algebra and Dynamics,” funded by the DFG (Project number 281071066 TRR 191). Research of the second author is partly supported by EPSRC Grant number EP/T015713/1 and partly by FWF Grant P-34501N.

Funding Information:
Research of the first author is partially supported by the DFG (Project Number 281071066 TRR 191). Research of the second author is partly supported by EPSRC Grant Number EP/T015713/1 and partly by FWF Grant P-34501N.

Publisher Copyright:
© 2022, The Author(s).

Keywords

Cylindric partitions
Partition diamonds
Partition identities
Skew double-shifted plane partitions
Weighted partition identities

ASJC Scopus subject areas

Algebra and Number Theory
Discrete Mathematics and Combinatorics

Access to Document

10.1007/s10801-022-01156-9Licence: CC BY

Cite this

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title = "Weighted cylindric 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{\"o}llnitz–Gordon and Little G{\"o}llnitz identities as well as some beautiful Schmidt-type identities of Andrews and Paule.",

keywords = "Cylindric partitions, Partition diamonds, Partition identities, Skew double-shifted plane partitions, Weighted partition identities",

author = "Walter Bridges and Uncu, {Ali K.}",

note = "Research of the first author is partially supported by the DFG (Project Number 281071066 TRR 191). Research of the second author is partly supported by EPSRC Grant Number EP/T015713/1 and partly by FWF Grant P-34501N. Funding Information: We would like to thank Krishnaswami Alladi, George E. Andrews, Jehanne Dousse, Guoniu Han, Ralf Hemmecke, and Peter Paule for their interest and encouragement. We would like to particularly thank Alexander Berkovich, Christian Krattencthaler, Ole Warnaar, and Wadim Zudilin for the discussion and their valuable feedback. We would also like to thank the anonymous referees for their careful reading and suggestions. The first author is partially supported by the SFB/TRR 191 “Symplectic Structures in Geometry, Algebra and Dynamics,” funded by the DFG (Project number 281071066 TRR 191). Research of the second author is partly supported by EPSRC Grant number EP/T015713/1 and partly by FWF Grant P-34501N. Funding Information: We would like to thank Krishnaswami Alladi, George E. Andrews, Jehanne Dousse, Guoniu Han, Ralf Hemmecke, and Peter Paule for their interest and encouragement. We would like to particularly thank Alexander Berkovich, Christian Krattencthaler, Ole Warnaar, and Wadim Zudilin for the discussion and their valuable feedback. We would also like to thank the anonymous referees for their careful reading and suggestions. The first author is partially supported by the SFB/TRR 191 “Symplectic Structures in Geometry, Algebra and Dynamics,” funded by the DFG (Project number 281071066 TRR 191). Research of the second author is partly supported by EPSRC Grant number EP/T015713/1 and partly by FWF Grant P-34501N. Funding Information: Research of the first author is partially supported by the DFG (Project Number 281071066 TRR 191). Research of the second author is partly supported by EPSRC Grant Number EP/T015713/1 and partly by FWF Grant P-34501N. Publisher Copyright: {\textcopyright} 2022, The Author(s).",

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month = dec,

doi = "10.1007/s10801-022-01156-9",

language = "English",

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AU - Uncu, Ali K.

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N2 - 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 Göllnitz–Gordon and Little Göllnitz identities as well as some beautiful Schmidt-type identities of Andrews and Paule.

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KW - Partition diamonds

KW - Partition identities

KW - Skew double-shifted plane partitions

KW - Weighted partition identities

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JO - Journal of Algebraic Combinatorics

JF - Journal of Algebraic Combinatorics

IS - 4

ER -

Weighted cylindric partitions

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Pushing Back the Doubly-Exponential Wall of Cylindrical Algebraic Decomposition

Cite this

Weighted cylindric partitions

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Projects

Pushing Back the Doubly-Exponential Wall of Cylindrical Algebraic Decomposition

Cite this