Partition Congruences by the Localization Method
Project Lead
Project Duration
01/03/2021 - 31/08/2023
Partners
The Austrian Science Fund (FWF)
Publications
2023
[Smoot]
A Congruence Family For 2-Elongated Plane Partitions: An Application of the Localization Method
N. Smoot
Journal of Number Theory 242, pp. 112-153. January 2023. ISSN 1096-1658. [doi]@article{RISC6661,
author = {N. Smoot},
title = {{A Congruence Family For 2-Elongated Plane Partitions: An Application of the Localization Method}},
language = {english},
abstract = {George Andrews and Peter Paule have recently conjectured an infinite family of congruences modulo powers of 3 for the 2-elongated plane partition function $d_2(n)$. This congruence family appears difficult to prove by classical methods. We prove a refined form of this conjecture by expressing the associated generating functions as elements of a ring of modular functions isomorphic to a localization of $mathbb{Z}[X]$.},
journal = {Journal of Number Theory},
volume = {242},
pages = {112--153},
isbn_issn = {ISSN 1096-1658},
year = {2023},
month = {January},
refereed = {yes},
length = {42},
url = {https://doi.org/10.1016/j.jnt.2022.07.014}
}
author = {N. Smoot},
title = {{A Congruence Family For 2-Elongated Plane Partitions: An Application of the Localization Method}},
language = {english},
abstract = {George Andrews and Peter Paule have recently conjectured an infinite family of congruences modulo powers of 3 for the 2-elongated plane partition function $d_2(n)$. This congruence family appears difficult to prove by classical methods. We prove a refined form of this conjecture by expressing the associated generating functions as elements of a ring of modular functions isomorphic to a localization of $mathbb{Z}[X]$.},
journal = {Journal of Number Theory},
volume = {242},
pages = {112--153},
isbn_issn = {ISSN 1096-1658},
year = {2023},
month = {January},
refereed = {yes},
length = {42},
url = {https://doi.org/10.1016/j.jnt.2022.07.014}
}
2022
[Sellers]
On the Divisibility of 7-Elongated Plane Partition Diamonds by Powers of 8
J. Sellers, N. Smoot
Technical report no. 22-17 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). February 2022. Licensed under CC BY 4.0 International. [doi] [pdf]@techreport{RISC6645,
author = {J. Sellers and N. Smoot},
title = {{On the Divisibility of 7-Elongated Plane Partition Diamonds by Powers of 8}},
language = {english},
abstract = {In 2021 da Silva, Hirschhorn, and Sellers studied a wide variety of congruences for the $k$-elongated plane partition function $d_k(n)$ by various primes. They also conjectured the existence of an infinite congruence family modulo arbitrarily high powers of 2 for the function $d_7(n)$. We prove that such a congruence family exists---indeed, for powers of 8. The proof utilizes only classical methods, i.e., integer polynomial manipulations in a single function, in contrast to all other known infinite congruence families for $d_k(n)$ which require more modern methods to prove.},
number = {22-17},
year = {2022},
month = {February},
keywords = {Partition congruences, infinite congruence family, modular functions, plane partitions, modular curve, Riemann surface},
length = {11},
license = {CC BY 4.0 International},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
author = {J. Sellers and N. Smoot},
title = {{On the Divisibility of 7-Elongated Plane Partition Diamonds by Powers of 8}},
language = {english},
abstract = {In 2021 da Silva, Hirschhorn, and Sellers studied a wide variety of congruences for the $k$-elongated plane partition function $d_k(n)$ by various primes. They also conjectured the existence of an infinite congruence family modulo arbitrarily high powers of 2 for the function $d_7(n)$. We prove that such a congruence family exists---indeed, for powers of 8. The proof utilizes only classical methods, i.e., integer polynomial manipulations in a single function, in contrast to all other known infinite congruence families for $d_k(n)$ which require more modern methods to prove.},
number = {22-17},
year = {2022},
month = {February},
keywords = {Partition congruences, infinite congruence family, modular functions, plane partitions, modular curve, Riemann surface},
length = {11},
license = {CC BY 4.0 International},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
[Smoot]
Divisibility Arising From Addition: The Application of Modular Functions to Infinite Partition Congruence Families
N. Smoot
Technical report no. 22-18 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). February 2022. Licensed under CC BY 4.0 International. [doi] [pdf]@techreport{RISC6659,
author = {N. Smoot},
title = {{Divisibility Arising From Addition: The Application of Modular Functions to Infinite Partition Congruence Families}},
language = {english},
abstract = {The theory of partition congruences has been a fascinating and difficult subject for over a century now. In attempting to prove a given congruence family, multiple possible complications include the genus of the underlying modular curve, representation difficulties of the associated sequences of modular functions, and difficulties regarding the piecewise $ell$-adic convergence of elements of the associated space of modular functions. However, our knowledge of the subject has developed substantially and continues to develop. In this very brief survey, we will discuss the utility of modular functions in proving partition congruences, both theoretical and computational, and many of the problems in the subject that are yet to be overcome.},
number = {22-18},
year = {2022},
month = {February},
keywords = {Partition congruences, modular functions, plane partitions, partition analysis, modular curve, Riemann surface},
length = {17},
license = {CC BY 4.0 International},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
author = {N. Smoot},
title = {{Divisibility Arising From Addition: The Application of Modular Functions to Infinite Partition Congruence Families}},
language = {english},
abstract = {The theory of partition congruences has been a fascinating and difficult subject for over a century now. In attempting to prove a given congruence family, multiple possible complications include the genus of the underlying modular curve, representation difficulties of the associated sequences of modular functions, and difficulties regarding the piecewise $ell$-adic convergence of elements of the associated space of modular functions. However, our knowledge of the subject has developed substantially and continues to develop. In this very brief survey, we will discuss the utility of modular functions in proving partition congruences, both theoretical and computational, and many of the problems in the subject that are yet to be overcome.},
number = {22-18},
year = {2022},
month = {February},
keywords = {Partition congruences, modular functions, plane partitions, partition analysis, modular curve, Riemann surface},
length = {17},
license = {CC BY 4.0 International},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
