arXiv subjects

James A. Sellers

Publications and source records attributed to James A. Sellers.

At least 37 records · Page 2Linked to original sources

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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arXiv2025

Arithmetic properties of MacMahon-type sums of divisors

In this paper, we prove several new infinite families of Ramanujan--like congruences satisfied by the coefficients of the generating function $U_t(a,q)$ which is an extension of MacMahon's generalized sum-of-divisors function. As a by-product, we also show that, for all $n\geq 0$, $\overline{B}_3(15n+7)\equiv 0 \pmod{5}$ where $\overline{B}_3(n)$ is the number of almost $3$-regular overpartitions of $n$.

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arXiv2025

Extending recent work of Nath, Saikia, and Sarma on $k$-tuple $\ell$-regular partitions

Let $T_{\ell,k}(n)$ denote the number of $\ell$-regular $k$-tuple partitions of $n$. In a recent work, Nath, Saikia, and Sarma derived several families of congruences for $T_{\ell,k}(n)$, with particular emphasis on the cases $T_{2,3}(n)$ and $T_{4,3}(n)$. In the concluding remarks of their paper, they conjectured that $T_{2,3}(n)$ satisfies an infinite set of congruences modulo 6. In this paper, we confirm their conjecture by proving a much more general result using elementary $q$-series techniques. We also present new families of congruences satisfied by $T_{\ell,k}(n)$.

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arXiv2024

Enumeration modulo 4 of overpartitions wherein only even parts can be overlined

In 2014, as part of a larger study of overpartitions with restrictions of the overlined parts based on residue classes, Munagi and Sellers defined $d_2(n)$ as the number of overpartitions of weight $n$ wherein only even parts can be overlined. As part of that work, they used a generating function approach to prove a parity characterization for $d_2(n)$. In this note, we give a combinatorial proof of their result and extend it to a modulus 4 characterization; we provide both generating function and combinatorial proofs of this stronger result. The combinatorial arguments incorporate classical involutions of Franklin, Glaisher, and Sylvester, along with a recent involution of van Leeuwen and methods new with this work.

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arXiv2024

Arithmetic properties modulo powers of $2$ for overpartition $k$-tuples with odd parts

Recently, Drema and Saikia (2023) proved several congruences modulo powers of 2 and 3 for overpartition triples with odd parts. We extend their list substantially. We prove several congruences modulo powers of 2 for overpartition k-tuples with odd parts, along with a few infinite families of congruences for overpartition triples with odd parts and for overpartition k-tuples with odd parts (for k = 4 and for odd k).

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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

Arithmetic properties for generalized cubic partitions and overpartitions modulo a prime

A cubic partition is an integer partition wherein the even parts can appear in two colors. In this paper, we introduce the notion of generalized cubic partitions and prove a number of new congruences akin to the classical Ramanujan-type. We emphasize two methods of proofs, one elementary (relying significantly on functional equations) and the other based on modular forms. We close by proving analogous results for generalized overcubic partitions.

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arXiv2024

d-Fold Partition Diamonds

In this work we introduce new combinatorial objects called $d$--fold partition diamonds, which generalize both the classical partition function and the partition diamonds of Andrews, Paule and Riese, and we set $r_d(n)$ to be their counting function. We also consider the Schmidt type $d$--fold partition diamonds, which have counting function $s_d(n).$ Using partition analysis, we then find the generating function for both, and connect the generating functions $\sum_{n= 0}^\infty s_d(n)q^n$ to Eulerian polynomials. This allows us to develop elementary proofs of infinitely many Ramanujan--like congruences satisfied by $s_d(n)$ for various values of $d$, including the following family: for all $d\geq 1$ and all $n\geq 0,$ $s_d(2n+1) \equiv 0 \pmod{2^d}.$

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arXiv2023

Elementary Proofs of Congruences for POND and PEND Partitions

Recently, Ballantine and Welch considered various generalizations and refinements of POD and PED partitions. 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). In the process, they were led to consider two classes of integer partitions which are, in some sense, the ``opposite'' of POD and PED partitions. They labeled these 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). In this work, we study these two types of partitions from an arithmetic perspective. Along the way, we are led to prove the following two infinite families of Ramanujan--like congruences: For all $α\geq 1$ and all $n\geq 0,$ \begin{align*} pond\left(3^{2α+1}n+\frac{23\cdot 3^{2α}+1}{8}\right) &\equiv 0 \pmod{3}, \textrm{\ \ \ and} \\ pend\left(3^{2α+1}n+\frac{17\cdot 3^{2α}-1}{8}\right) &\equiv 0 \pmod{3} \end{align*} where $pond(n)$ counts the number of POND partitions of weight $n$ and $pend(n)$ counts the number of PEND partitions of weight $n$. All of the proof techniques used herein are elementary, relying on classical $q$-series identities and generating function manipulations, along with mathematical induction.

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arXiv2023

Elementary Proofs of Arithmetic Properties for Schur-Type Overpartitions Modulo Small Powers of 2

In 2022, Broudy and Lovejoy extensively studied the function $S(n)$ which counts the number of overpartitions of \emph{Schur-type}. In particular, they proved a number of congruences satisfied by $S(n)$ modulo $2$, $4$, and $5$. In this work, we extend their list of arithmetic properties satisfied by $S(n)$ by focusing on moduli which are small powers of 2. In particular, we prove the following infinite family of Ramanujan-like congruences: For all $α\geq 0$ and $n\geq 0$, $$ S\left(2^{5+2α}n+\left(2^{5+2α}-\frac{2^{2+2α}-1}{3}\right)\right)\equiv 0 \pmod{16}. $$ All of the proof techniques used herein are elementary, relying on classical $q$-series identities and generating function manipulations as well as the parameterization work popularized by Alaca, Alaca, and Williams.

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arXiv2023

An infinite family of internal congruences modulo powers of 2 for partitions into odd parts with designated summands

In 2002, Andrews, Lewis, and Lovejoy introduced the combinatorial objects which they called \emph{partitions with designated summands}. These are built by taking unrestricted integer partitions and designating exactly one of each occurrence of a part. In that same work, Andrews, Lewis, and Lovejoy also studied such partitions wherein all parts must be odd, and they denoted the number of such partitions of size $n$ by the function $PDO(n)$. Since then, numerous authors have proven a variety of divisibility properties satisfied by $PDO(n)$. Recently, the second author proved the following internal congruences satisfied by $PDO(n)$: For all $n\geq 0$, \begin{align*} PDO(4n) &\equiv PDO(n) \pmod{4},\\ PDO(16n) &\equiv PDO(4n) \pmod{8}. \end{align*} In this work, we significantly extend these internal congruence results by proving the following new infinite family of congruences: For all $k\geq 0$ and all $n\geq 0$, $$PDO(2^{2k+3}n) \equiv PDO(2^{2k+1}n) \pmod{2^{2k+3}}.$$ We utilize several classical tools to prove this family, including generating function dissections via the unitizing operator of degree two, various modular relations and recurrences involving a Hauptmodul on the classical modular curve $X_0(6)$, and an induction argument which provides the final step in proving the necessary divisibilities. It is notable that the construction of each $2$-dissection slice of our generating function bears an entirely different nature to those studied in the past literature.

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arXiv2023

An Elementary Proof of a Conjecture of Saikia on Congruences for $t$--Colored Overpartitions

The starting point for this work is the family of functions $\overline{p}_{-t}(n)$ which counts the number of $t$--colored overpartitions of $n.$ In recent years, several infinite families of congruences satisfied by $\overline{p}_{-t}(n)$ for specific values of $t\geq 1$ have been proven. In particular, in his 2023 work, Saikia proved a number of congruence properties modulo powers of 2 for $\overline{p}_{-t}(n)$ for $t=5,7,11,13$. He also included the following conjecture in that paper: \newline \ %\newline \noindent Conjecture: For all $n\geq 0$ and primes $t$, we have \begin{eqnarray*} \overline{p}_{-t}(8n+1) &\equiv & 0 \pmod{2}, \\ \overline{p}_{-t}(8n+2) &\equiv & 0 \pmod{4}, \\ \overline{p}_{-t}(8n+3) &\equiv & 0 \pmod{8}, \\ \overline{p}_{-t}(8n+4) &\equiv & 0 \pmod{2}, \\ \overline{p}_{-t}(8n+5) &\equiv & 0 \pmod{8}, \\ \overline{p}_{-t}(8n+6) &\equiv & 0 \pmod{8}, \\ \overline{p}_{-t}(8n+7) &\equiv & 0 \pmod{32}. \end{eqnarray*} Using a truly elementary approach, relying on classical generating function manipulations and dissections, as well as proof by induction, we show that Saikia's conjecture holds for {\bf all} odd integers $t$ (not necessarily prime).

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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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arXiv2023

New Infinite Families of Congruences Modulo Powers of 2 for 2--Regular Partitions with Designated Summands

In 2002, Andrews, Lewis, and Lovejoy introduced the combinatorial objects which they called {\it partitions with designated summands}. These are built by taking unrestricted integer partitions and designating exactly one of each occurrence of a part. In that same work, Andrews, Lewis, and Lovejoy also studied such partitions wherein all parts must be odd. Recently, Herden, Sepanski, Stanfill, Hammon, Henningsen, Ickes, and Ruiz proved a number of Ramanujan--like congruences for the function $PD_2(n)$ which counts the number of partitions of weight $n$ with designated summands wherein all parts must be odd. In this work, we prove some of the results conjectured by Herden, et. al. by proving the following two infinite families of congruences satisfied by $PD_2(n)$: For all $α\geq 0$ and $n\geq 0,$ \begin{eqnarray*} PD_2(2^α(4n+3)) &\equiv & 0 \pmod{4} \ \ \ \ \ {\textrm and} \\ PD_2(2^α(8n+7)) &\equiv & 0 \pmod{8}. \end{eqnarray*} All of the proof techniques used herein are elementary, relying on classical $q$--series identities and generating function manipulations.

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arXiv2023

On Blecher and Knopfmacher's Fixed Points for Integer Partitions

Recently, Blecher and Knopfmacher explored the notion of fixed points in integer partitions and hypothesized on the relative number of partitions with and without a fixed point. We resolve their open question by working fixed points into a growing number of interconnected partition statistics involving Frobenius symbols, Dyson's crank, and the mex (minimal excluded part). Also, we generalize the definition of fixed points and connect that expanded notion to the $\text{mex}_j$ defined by Hopkins, Sellers, and Stanton as well as the $j$-Durfee rectangle defined by Hopkins, Sellers, and Yee.

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arXiv2022

Elementary Proofs of Infinite Families of Congruences for Merca's Cubic Partitions

Recently, using modular forms and Smoot's {\tt Mathematica} implementation of Radu's algorithm for proving partition congruences, Merca proved the following two congruences: For all $n\geq 0,$ \begin{align*} A(9n+5) & \equiv 0 \pmod{3}, \\ A(27n+26) & \equiv 0 \pmod{3}. \end{align*} Here $A(n)$ is closely related to the function which counts the number of {\it cubic partitions}, partitions wherein the even parts are allowed to appear in two different colors. Indeed, $A(n)$ is defined as the difference between the number of cubic partitions of $n$ into an even numbers of parts and the number of cubic partitions of $n$ into an odd numbers of parts. In this brief note, we provide elementary proofs of these two congruences via classical generating function manipulations. We then prove two infinite families of non--nested Ramanujan--like congruences modulo 3 satisfied by $A(n)$ wherein Merca's original two congruences serve as the initial members of each family.

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arXiv2022

On the Parity of the Generalized Frobenius Partition Functions $ϕ_k(n)$

In his 1984 Memoir of the American Mathematical Society, George Andrews defined two families of functions, $ϕ_k(n)$ and $cϕ_k(n),$ which enumerate two types of combinatorial objects which Andrews called generalized Frobenius partitions. As part of that Memoir, Andrews proved a number of Ramanujan--like congruences satisfied by specific functions within these two families. In the years that followed, numerous other authors proved similar results for these functions, often with a view towards a specific choice of the parameter $k.$ In this brief note, our goal is to identify an {\bf infinite} family of values of $k$ such that $ϕ_k(n)$ is even for all $n$ in a specific arithmetic progression; in particular, our primary goal in this work is to prove that, for all positive integers $\ell,$ all primes $p\geq 5,$ and all values $r,$ $0 < r < p,$ such that $24r+1$ is a quadratic nonresidue modulo $p,$ $$ ϕ_{p\ell-1}(pn+r) \equiv 0 \pmod{2} $$ for all $n\geq 0.$ Our proof of this result is truly elementary, relying on a lemma from Andrews' Memoir, classical $q$--series results, and elementary generating function manipulations. Such a result, which holds for infinitely many values of $k,$ is rare in the study of arithmetic properties satisfied by generalized Frobenius partitions, primarily because of the unwieldy nature of the generating functions in question.

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arXiv2021

Elementary Proofs of Infinitely Many Congruences for $k$-Elongated Partition Diamonds

In 2007, Andrews and Paule published the eleventh paper in their series on MacMahon's partition analysis, with a particular focus on broken $k$-diamond partitions. On the way to broken $k$-diamond partitions, Andrews and Paule introduced the idea of $k$-elongated partition diamonds. Recently, Andrews and Paule revisited the topic of $k$-elongated partition diamonds. Using partition analysis and the Omega operator, they proved that the generating function for the partition numbers $d_k(n)$ produced by summing the links of $k$-elongated plane partition diamonds of length $n$ is given by $\frac{(q^2;q^2)_\infty^k}{(q;q)_\infty^{3k+1}}$ for each $k\geq 1.$ A significant portion of their recent paper involves proving several congruence properties satisfied by $d_1, d_2$ and $d_3$, using modular forms as their primary proof tool. In this work, our goal is to extend some of the results proven by Andrews and Paule in their recent paper by proving infinitely many congruence properties satisfied by the functions $d_k$ for an infinite set of values of $k.$ The proof techniques employed are all elementary, relying on generating function manipulations and classical $q$-series results.

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