RaduRK is a Mathematica implementation of an algorithm developed by Cristian-Silviu Radu. The algorithm takes as input an arithmetic sequence a(n) generated from a large class of q-Pochhammer quotients, together with a given arithmetic progression mn+j, and the level of ...

# Dr. Nicolas Smoot

### Research Area

Analytic Number Theory, Modular Forms, Partition Theory## Software

Authors: Nicolas Smoot

More## 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]@

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}

}

**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}

}

[Smoot]

### 2-Elongated Plane Partitions and Powers of 7: The Localization Method Applied to a Genus 1 Congruence Family

#### K. Banerjee, N.A. Smoot

Technical report no. 23-10 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). August 2023. Licensed under CC BY 4.0 International. [doi] [pdf]@

author = {K. Banerjee and N.A. Smoot},

title = {{2-Elongated Plane Partitions and Powers of 7: The Localization Method Applied to a Genus 1 Congruence Family}},

language = {english},

abstract = {Over the last century, a large variety of infinite congruence families have been discovered and studied, exhibiting a great variety with respect to their difficulty. Major complicating factors arise from the topology of the associated modular curve: classical techniques are sufficient when the associated curve has cusp count 2 and genus 0. Recent work has led to new techniques that have proven useful when the associated curve has cusp count greater than 2 and genus 0. We show here that these techniques may be adapted in the case of positive genus. In particular, we examine a congruence family over the 2-elongated plane partition diamond counting function $d_2(n)$ by powers of 7, for which the associated modular curve has cusp count 4 and genus 1. We compare our method with other techniques for proving genus 1 congruence families, and present a second congruence family by powers of 7 which we conjecture, and which may be amenable to similar techniques.},

number = {23-10},

year = {2023},

month = {August},

keywords = {Partition congruences, infinite congruence family, modular functions, plane partitions, partition analysis, modular curve, Riemann surface},

length = {35},

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)}

}

**techreport**{RISC6713,author = {K. Banerjee and N.A. Smoot},

title = {{2-Elongated Plane Partitions and Powers of 7: The Localization Method Applied to a Genus 1 Congruence Family}},

language = {english},

abstract = {Over the last century, a large variety of infinite congruence families have been discovered and studied, exhibiting a great variety with respect to their difficulty. Major complicating factors arise from the topology of the associated modular curve: classical techniques are sufficient when the associated curve has cusp count 2 and genus 0. Recent work has led to new techniques that have proven useful when the associated curve has cusp count greater than 2 and genus 0. We show here that these techniques may be adapted in the case of positive genus. In particular, we examine a congruence family over the 2-elongated plane partition diamond counting function $d_2(n)$ by powers of 7, for which the associated modular curve has cusp count 4 and genus 1. We compare our method with other techniques for proving genus 1 congruence families, and present a second congruence family by powers of 7 which we conjecture, and which may be amenable to similar techniques.},

number = {23-10},

year = {2023},

month = {August},

keywords = {Partition congruences, infinite congruence family, modular functions, plane partitions, partition analysis, modular curve, Riemann surface},

length = {35},

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)}

}

### 2022

[Smoot]

### 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]@

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)}

}

**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)}

}

[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]@

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)}

}

**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)}

}

[Smoot]

### The localization method applied to k-elongated plane partitions and divisibily by 5

#### K. Banerjee, N. A. Smoot

Technical report no. 22-21 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). August 2022. Licensed under CC BY 4.0 International. [doi] [pdf]@

author = {K. Banerjee and N. A. Smoot},

title = {{The localization method applied to k-elongated plane partitions and divisibily by 5}},

language = {english},

abstract = {The enumeration $d_k(n)$ of k-elongated plane partition diamonds has emerged as a generalization of the classical integer partition function p(n). We have discovered an infinite congruence family for $d_5(n)$ modulo powers of 5. Classical methods cannot be used to prove this family of congruences. Indeed, the proof employs the recently developed localization method, and utilizes a striking internal algebraic structure which has not yet been seen in the proof of any congruence family. We believe that this discovery poses important implications on future work in partition congruences.},

number = {22-21},

year = {2022},

month = {August},

keywords = {Partition congruences, modular functions, plane partitions, partition analysis, Ramanujan’s theta functions, localization method, modular curve, Riemann surface},

length = {40},

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)}

}

**techreport**{RISC6725,author = {K. Banerjee and N. A. Smoot},

title = {{The localization method applied to k-elongated plane partitions and divisibily by 5}},

language = {english},

abstract = {The enumeration $d_k(n)$ of k-elongated plane partition diamonds has emerged as a generalization of the classical integer partition function p(n). We have discovered an infinite congruence family for $d_5(n)$ modulo powers of 5. Classical methods cannot be used to prove this family of congruences. Indeed, the proof employs the recently developed localization method, and utilizes a striking internal algebraic structure which has not yet been seen in the proof of any congruence family. We believe that this discovery poses important implications on future work in partition congruences.},

number = {22-21},

year = {2022},

month = {August},

keywords = {Partition congruences, modular functions, plane partitions, partition analysis, Ramanujan’s theta functions, localization method, modular curve, Riemann surface},

length = {40},

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)}

}

### 2021

[Smoot]

### A method of verifying partition congruences by symbolic computation

#### Nicolas Allen Smoot, Cristian-Silviu Radu

Journal of Symbolic Computation 104, pp. 105-133. 2021. 0747-7171. [doi] [pdf]@

author = {Nicolas Allen Smoot and Cristian-Silviu Radu},

title = {{A method of verifying partition congruences by symbolic computation}},

language = {english},

abstract = {Conjectures involving infinite families of restricted partition congruences can be difficult to verify for a number of individual cases, even with a computer. We demonstrate how the machinery of Radu's algorithm may be modified and employed to efficiently check a very large number of cases of such conjectures. This allows substantial evidence to be collected for a given conjecture, before a complete proof is attempted.},

journal = {Journal of Symbolic Computation},

volume = {104},

pages = {105--133},

isbn_issn = {0747-7171},

year = {2021},

refereed = {yes},

length = {29},

url = {https://doi.org/10.1016/j.jsc.2020.04.008}

}

**article**{RISC6106,author = {Nicolas Allen Smoot and Cristian-Silviu Radu},

title = {{A method of verifying partition congruences by symbolic computation}},

language = {english},

abstract = {Conjectures involving infinite families of restricted partition congruences can be difficult to verify for a number of individual cases, even with a computer. We demonstrate how the machinery of Radu's algorithm may be modified and employed to efficiently check a very large number of cases of such conjectures. This allows substantial evidence to be collected for a given conjecture, before a complete proof is attempted.},

journal = {Journal of Symbolic Computation},

volume = {104},

pages = {105--133},

isbn_issn = {0747-7171},

year = {2021},

refereed = {yes},

length = {29},

url = {https://doi.org/10.1016/j.jsc.2020.04.008}

}

### 2020

[Smoot]

### On the Computation of Identities Relating Partition Numbers in Arithmetic Progressions with Eta Quotients: An Implementation of Radu's Algorithm

#### Nicolas Allen Smoot

Journal of Symbolic Computation, pp. -. 2020. 0747-7171. To Appear. [pdf]@

author = {Nicolas Allen Smoot},

title = {{On the Computation of Identities Relating Partition Numbers in Arithmetic Progressions with Eta Quotients: An Implementation of Radu's Algorithm}},

language = {english},

abstract = {In 2015 Cristian-Silviu Radu designed an algorithm to detect identities of a class studied by Ramanujan and Kolberg, in which the generating functions of a partition function over a given set of arithmetic progression are expressed in terms of Dedekind eta quotients over a given congruence subgroup. These identities include the famous results by Ramanujan which provide a witness to the divisibility properties of $p(5n+4),$ $p(7n+5)$. We give an implementation of this algorithm using Mathematica. The basic theory is first described, and an outline of the algorithm is briefly given, in order to describe the functionality and utility of our package. We thereafter give multiple examples of applications to recent work in partition theory. In many cases we have used our package to derive alternate proofs of various identities or congruences; in other cases we have improved previously established identities.},

journal = {Journal of Symbolic Computation},

pages = {--},

isbn_issn = {0747-7171},

year = {2020},

note = {To Appear},

refereed = {yes},

length = {32}

}

**article**{RISC6105,author = {Nicolas Allen Smoot},

title = {{On the Computation of Identities Relating Partition Numbers in Arithmetic Progressions with Eta Quotients: An Implementation of Radu's Algorithm}},

language = {english},

abstract = {In 2015 Cristian-Silviu Radu designed an algorithm to detect identities of a class studied by Ramanujan and Kolberg, in which the generating functions of a partition function over a given set of arithmetic progression are expressed in terms of Dedekind eta quotients over a given congruence subgroup. These identities include the famous results by Ramanujan which provide a witness to the divisibility properties of $p(5n+4),$ $p(7n+5)$. We give an implementation of this algorithm using Mathematica. The basic theory is first described, and an outline of the algorithm is briefly given, in order to describe the functionality and utility of our package. We thereafter give multiple examples of applications to recent work in partition theory. In many cases we have used our package to derive alternate proofs of various identities or congruences; in other cases we have improved previously established identities.},

journal = {Journal of Symbolic Computation},

pages = {--},

isbn_issn = {0747-7171},

year = {2020},

note = {To Appear},

refereed = {yes},

length = {32}

}

[Smoot]

### A Single-Variable Proof of the Omega SPT Congruence Family Over Powers of 5

#### Nicolas Allen Smoot

Research Institute for Symbolic Computation. Technical report, 2020. Submitted. [pdf]@

author = {Nicolas Allen Smoot},

title = {{A Single-Variable Proof of the Omega SPT Congruence Family Over Powers of 5}},

language = {english},

abstract = {In 2018 Liuquan Wang and Yifan Yang proved the existence of an infinite family of congruences for the smallest parts function corresponding to the third order mock theta function $\omega(q)$. Their proof took the form of an induction requiring 20 initial relations, and utilized a space of modular functions isomorphic to a free rank 2 $\mathbb{Z}[X]$-module. This proof strategy was originally developed by Paule and Radu to study families of congruences associated with modular curves of genus 1. We show that Wang and Yang's family of congruences, which is associated with a genus 0 modular curve, can be proved using a single-variable approach, via a ring of modular functions isomorphic to a localization of $\mathbb{Z}[X]$. To our knowledge, this is the first time that such an algebraic structure has been applied to the theory of partition congruences. Our induction is more complicated, and relies on sequences of functions which exhibit a somewhat irregular 5-adic growth. However, the proof ultimately rests upon the direct verification of only 10 initial relations, and is similar to the classical methods of Ramanujan and Watson.},

year = {2020},

note = {Submitted},

institution = {Research Institute for Symbolic Computation},

length = {32}

}

**techreport**{RISC6107,author = {Nicolas Allen Smoot},

title = {{A Single-Variable Proof of the Omega SPT Congruence Family Over Powers of 5}},

language = {english},

abstract = {In 2018 Liuquan Wang and Yifan Yang proved the existence of an infinite family of congruences for the smallest parts function corresponding to the third order mock theta function $\omega(q)$. Their proof took the form of an induction requiring 20 initial relations, and utilized a space of modular functions isomorphic to a free rank 2 $\mathbb{Z}[X]$-module. This proof strategy was originally developed by Paule and Radu to study families of congruences associated with modular curves of genus 1. We show that Wang and Yang's family of congruences, which is associated with a genus 0 modular curve, can be proved using a single-variable approach, via a ring of modular functions isomorphic to a localization of $\mathbb{Z}[X]$. To our knowledge, this is the first time that such an algebraic structure has been applied to the theory of partition congruences. Our induction is more complicated, and relies on sequences of functions which exhibit a somewhat irregular 5-adic growth. However, the proof ultimately rests upon the direct verification of only 10 initial relations, and is similar to the classical methods of Ramanujan and Watson.},

year = {2020},

note = {Submitted},

institution = {Research Institute for Symbolic Computation},

length = {32}

}

[Smoot]

### Computer Algebra with the Fifth Operation: Applications of Modular Functions to Partition Congruences

#### N. Smoot

Research Institute for Symbolic Computation, JKU Linz. PhD Thesis. 2020. [pdf]@

author = {N. Smoot},

title = {{Computer Algebra with the Fifth Operation: Applications of Modular Functions to Partition Congruences}},

language = {english},

abstract = {We give the implementation of an algorithm developed by Silviu Radu to compute examples of a wide variety of arithmetic identities originally studied by Ramanujan and Kolberg. Such identities employ certain finiteness conditions imposed by the theory of modular functions, and often yield interesting arithmetic information about the integer partition function $p(n)$, and other associated functions. We compute a large number of examples of such identities taken from contemporary research, often extending or improving existing results. We then use our implementation as a computational tool to help us achieve more theoretical results in the study of infinite congruence families. We finally describe a new method which extends the existing techniques for proving partition congruence families associated with a genus 0 modular curve.},

year = {2020},

translation = {0},

school = {Research Institute for Symbolic Computation, JKU Linz},

length = {224}

}

**phdthesis**{RISC6512,author = {N. Smoot},

title = {{Computer Algebra with the Fifth Operation: Applications of Modular Functions to Partition Congruences}},

language = {english},

abstract = {We give the implementation of an algorithm developed by Silviu Radu to compute examples of a wide variety of arithmetic identities originally studied by Ramanujan and Kolberg. Such identities employ certain finiteness conditions imposed by the theory of modular functions, and often yield interesting arithmetic information about the integer partition function $p(n)$, and other associated functions. We compute a large number of examples of such identities taken from contemporary research, often extending or improving existing results. We then use our implementation as a computational tool to help us achieve more theoretical results in the study of infinite congruence families. We finally describe a new method which extends the existing techniques for proving partition congruence families associated with a genus 0 modular curve.},

year = {2020},

translation = {0},

school = {Research Institute for Symbolic Computation, JKU Linz},

length = {224}

}

### 2019

[Smoot]

### A Family of Congruences for Rogers-Ramanujan Subpartitions

#### Nicolas Allen Smoot

Journal of Number Theory 196, pp. 35-60. March 2019. ISSN 0022-314X. [pdf]@

author = {Nicolas Allen Smoot},

title = {{A Family of Congruences for Rogers--Ramanujan Subpartitions}},

language = {english},

abstract = {In 2015 Choi, Kim, and Lovejoy studied a weighted partition function, A1(m), which counted subpartitions with a structure related to the Rogers–Ramanujan identities. They conjectured the existence of an infinite class of congruences for A1(m), modulo powers of 5. We give an explicit form of this conjecture, and prove it for all powers of 5.},

journal = {Journal of Number Theory},

volume = {196},

pages = {35--60},

isbn_issn = {ISSN 0022-314X},

year = {2019},

month = {March},

refereed = {yes},

keywords = {Integer partitions, Partition congruences, Rogers--Ramanujan identities, Ramanujan--Kolberg identities, Modular functions},

sponsor = {FWF: W1214-N15},

length = {26}

}

**article**{RISC5809,author = {Nicolas Allen Smoot},

title = {{A Family of Congruences for Rogers--Ramanujan Subpartitions}},

language = {english},

abstract = {In 2015 Choi, Kim, and Lovejoy studied a weighted partition function, A1(m), which counted subpartitions with a structure related to the Rogers–Ramanujan identities. They conjectured the existence of an infinite class of congruences for A1(m), modulo powers of 5. We give an explicit form of this conjecture, and prove it for all powers of 5.},

journal = {Journal of Number Theory},

volume = {196},

pages = {35--60},

isbn_issn = {ISSN 0022-314X},

year = {2019},

month = {March},

refereed = {yes},

keywords = {Integer partitions, Partition congruences, Rogers--Ramanujan identities, Ramanujan--Kolberg identities, Modular functions},

sponsor = {FWF: W1214-N15},

length = {26}

}