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Nicolas Allen Smoot

Publications and source records attributed to Nicolas Allen Smoot.

15 recordsLinked to original sources
arXiv2026

On the occurrence of congruence multiplicities between Ramanujan's theta functions

Recently, the first and third authors initiated a study of arithmetical relationships between Ramanujan's theta functions $\varphi(-q)$ and $\psi(q)$. In this work, we prove eight families of internal congruences modulo arbitrary powers of $3$ and $5$ for infinite series related to the two theta functions. We also show that there exist isomorphisms between the congruence families for $\varphi(-q)$ and $\psi(q)$, which can be realized by the study of congruence multiplicities previously studied by Garvan, Sellers, and the second author. We believe that such equivalences are exclusive, at least on the congruence subgroups $\Gamma_0(6)$ and $\Gamma_0(10)$. In the end, we show how a simple manipulation of function field extensions allows us to predict whether additional isomorphisms to our congruences occur.

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arXiv2025

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

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 conjecture a second congruence family by powers of 7, which may be amenable to similar techniques.

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arXiv2025

Explaining Unforeseen Congruence Relationships Between PEND and POND Partitions via an Atkin--Lehner Involution

For the past several years, numerous authors have studied POD and PED partitions from a variety of perspectives. These are integer partitions wherein the odd parts must be distinct (in the case of POD partitions) or the even parts must be distinct (in the case of PED partitions). More recently, Ballantine and Welch were led to consider POND and PEND partitions, which are integer partitions wherein the odd parts cannot be distinct (in the case of POND partitions) or the even parts cannot be distinct (in the case of PEND partitions). Soon after, the first author proved the following results via elementary $q$-series identities and generating function manipulations, along with mathematical induction: For all $α\geq 1$ and all $n\geq 0,$ $$\mathrm{pend}\left(3^{2α+1}n+\frac{17\cdot 3^{2α}-1}{8}\right) \equiv 0 \pmod{3}, \textrm{ and}$$ $$\mathrm{pond}\left(3^{2α+1}n+\frac{23\cdot 3^{2α}+1}{8}\right) \equiv 0 \pmod{3},$$ where $\mathrm{pend}(n)$ counts the number of PEND partitions of weight $n$ and $\mathrm{pond}(n)$ counts the number of POND partitions of weight $n$. In this work, we revisit these families of congruences, and we show a relationship between them via an Atkin--Lehner involution. From this relationship, we can show that, once one of the above families of congruences is known, the other follows immediately.

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arXiv2024

The Localization Method Applied to $k$-Elongated Plane Partitions and Divisibility by 5

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.

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arXiv2024

Old Meets New: Connecting Two Infinite Families of Congruences Modulo Powers of 5 for Generalized Frobenius Partition Functions

In 2012 Paule and Radu proved a difficult family of congruences modulo powers of 5 for Andrews' 2-colored generalized Frobenius partition function. The family is associated with the classical modular curve of level 20. We demonstrate the existence of a congruence family for a related generalized Frobenius partition function associated with the same curve. We construct an isomorphism between this new family and the original family of congruences via a mapping on the associated rings of modular functions. The pairing of the congruence families provides a new strategy for future work on congruences associated with modular curves of composite level. We show how a similar approach can be made to multiple other recent examples in the literature. We also give some important insights into the behavior of these congruence families with respect to the Atkin--Lehner involution which proved very important in Paule and Radu's original proof.

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arXiv2024

On the Classification of Modular Congruence Families

Congruence families, i.e., $\ell$-adic convergence for well-defined arithmetic subsequences, is a commonplace phenomenon for the coefficients of modular forms. Such families superficially resemble one another, but they often vary substantially in difficulty. Moreover, the critical difficulties associated with a given family will generally manifest themselves at the end stages of an attempted proof. We give a conjectured classification system of congruence families for the coefficients of modular eta quotients by studying the topology of the associated modular curve.

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arXiv2023

On the Divisibility of 7-Elongated Plane Partition Diamonds by Powers of 8

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.

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arXiv2022

Divisibility Arising From Addition: The Application of Modular Functions to Infinite Partition Congruence Families

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.

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arXiv2022

A Congruence Family For 2-Elongated Plane Partitions: An Application of the Localization Method

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 $\mathbb{Z}[X]$.

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arXiv2020

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

In 2015 Cristian-Silviu Radu designed an algorithm to detect identities of a class studied by Ramanujan and Kolberg. This class includes the famous identities 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, and in at least one case we have confirmed a standing conjecture.

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arXiv2020

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

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 $ω(q)$. 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.

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arXiv2020

A Family of Congruences for Rogers--Ramanujan Subpartitions

In 2015 Choi, Kim, and Lovejoy studied a weighted partition function, $A_1(m)$, which counted subpartitions with a structure related to the Rogers--Ramanujan identities. They conjectured the existence of an infinite class of congruences for $A_1(m)$, modulo powers of 5. We give an explicit form of this conjecture, and prove it for all powers of 5.

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arXiv2020

A Method of Verifying Partition Congruences by Symbolic Computation

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.

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