arXiv subjects

James A. Sellers

Publications and source records attributed to James A. Sellers.

At least 19 recordsLinked to original sources

arXiv2026

Congruences for Overcubic Partition $k$-Tuples

In the last few years, a number of authors have proved divisibility properties satisfied by various functions which count the number of overcubic partition $k$--tuples of weight $n$ for small values of $k$. In this work, we use generating functions to prove some of their results as well as multiple infinite families of new congruences for overcubic partition $k$-tuples which do not yet appear in the literature. In particular, we focus on a new perspective which provides insights as to why these functions are often divisible by powers of 2, and we also prove families of congruences whose moduli are odd. For example, we prove that, for all $m\geq 0$, $\OL{b}_{4}(22m + 11) \equiv 0 \pmod{11}$ and we also prove infinite families such as $\OL{b}_{9l+2}(9m+3) \equiv 0 \pmod{3}$ for all $m,l\geq0$.

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arXiv2026

A uniform proof approach for congruences modulo 3 for partitions with $k$-colored odd parts

In recent work, Hirschhorn and the second author defined $a_k(n)$ to be the number of partitions of $n$ wherein the even parts come in only one color, while the odd parts may be ``colored'' with one of $k$ colors for fixed $k\geq 1$. This function generalizes the classical partition function and has been of significant interest because it satisfies a number of congruences. Although prior work studying $a_k(n)$ has resulted in congruences in arithmetic progressions in a somewhat ad hoc manner, this work gives a uniform framework for studying congruences modulo 3. This allows us to prove an infinite family of infinite families of non-nested congruences modulo 3.

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arXiv2026

Elementary proofs of congruences modulo 5 for overpartitions with restricted odd differences

In 2015, Bringmann, Dousse, Lovejoy, and Mahlburg defined the function $\overline{t}(n)$ to be the number of overpartitions of weight $n$ where (i) the difference between two successive parts may be odd only if the larger part is overlined and (ii) if the smallest part is odd then it is overlined. In their work, they proved that $\overline{t}(n)$ satisfies an elegant congruence modulo 3. Since then, a number of authors have studied arithmetic properties satisfied by $\overline{t}(n)$. In particular, in 2023, Hanson and Smith utilized the theory of modular forms to prove the following two congruences modulo 5: For all $n\geq 0$, \begin{align*} \overline{t}(80n+40)\equiv \overline{t}(80n+60)\equiv 0 \pmod{5}. \end{align*} Our goal in this work is to provide a truly elementary proof of this pair of congruences.

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arXiv2026

Eta-Quotient Representations for a Three Parameter Family of Modular Functions Associated with the Rogers-Ramanujan Continued Fraction

In 2021, Chern and Tang introduced two families of two-parameter modular functions associated with the Rogers-Ramanujan continued fraction. They established recurrence relations that express the members of these families in terms of eta quotients, and used these expressions to obtain dissection formulas. Motivated by their work, we construct a three-parameter extension and derive the corresponding recursive eta quotient representations. Our result also has applications to dissection formulas. In particular, it is used in separate work to obtain $5$-dissection formulas for overpartitions with restricted odd differences.

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arXiv2026

Convolutive sequences, II: Parametrizations

In recent work, the authors defined a sequence $(a_n)_{n\ge 0}$ to be $m$-convolutive exactly if \begin{align*} \sum_{n\ge 0} a_{mn} q^n = \left(\sum_{n\ge 0} a_n q^n\right)^m \end{align*} for a specific positive integer $m$ and provided proofs of the $2$- and $3$-convolutivity of a small number of sequences arising from primitive eta-products. Since the completion of that work, the authors have discovered many new instances of convolutive eta-products. The main focus of this work is to unify all but one of these instances in a parametric way.

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arXiv2026

Arithmetic Properties Satisfied by a Recent Integer Partition Function of Dombos

In recent work of Dombos, the set of integer partitions of $n$ wherein the parts are either divisible by 4 or congruent to $\pm 1 \pmod{6}$ arose in a natural way. In this work, we will denote the function which counts the number of such partitions of $n$ by $dp(n)$. Using elementary generating function manipulations and classical $q$--series results, we prove several congruences satisfied by $dp(n)$. As an example, we prove that, for all $α\geq 1$ and all $n \geq 0$, \begin{equation*} dp \left( 3^{2α+ 1}n + \frac{7 \cdot 9^α+ 1}{4} \right) \equiv 0 \pmod{3}. \end{equation*}

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arXiv2026

New arithmetic properties for overpartitions where nonoverlined parts are $\ell$-regular

In this paper, we study the partition functions $\overline{R_\ell^\ast}(n)$, which count the number of overpartitions of $n$ where the non-overlined parts are $\ell$-regular for a given $\ell$. Using elementary techniques, as well as the theory of modular forms, we establish several new arithmetic properties, including infinite families of congruences for these functions.

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arXiv2026

A Family of Congruences Modulo 7 for Partitions with Monochromatic Even Parts and Multi--Colored Odd Parts

In recent work, Amdeberhan and Merca considered the integer partition function $a(n)$ which counts the number of integer partitions of weight $n$ wherein even parts come in only one color (i.e., they are monochromatic), while the odd parts may appear in one of three colors. One of the results that they proved was that, for all $n\geq 0$, $a(7n+2) \equiv 0 \pmod{7}$. In this work, we generalize this function $a(n)$ by naturally placing it within an infinite family of related partition functions. Using elementary generating function manipulations and classical $q$--series identities, we then prove infinitely many congruences modulo 7 which are satisfied by members of this family of functions.

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arXiv2026

Arithmetic Properties of Colored Partitions Restricted by Parity of the Parts

Let $a_{r,s}(n)$ denote the number of mutlicolored partitions of $n$, wherein both even parts and odd parts may appear in one of $r$-colors and $s$-colors, respectively, for fixed $r,s\ge 1$. The paper aims to study arithmetic properties satisfied by $a_{r,s}(n)$, using elementary generating function manipulations and classical $q$-series techniques.

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arXiv2026

Additional Congruences for generalized Color Partitions of Hirschhorn and Sellers

Let $a_k(n)$ denote the number of partitions of $n$ wherein even parts come in only one color, while the odd parts may be ``colored" with one of $k$ colors, for fixed $k$. In this note, we find some congruences for $a_k(n)$ in the spirit of Ramanujan's congruences. We prove a number of results for $a_k(n)$ modulo powers of $2$ for infinitely many values of $k$. Our approach is truly elementary, relying on generating function manipulations, theta functions and $q$-dissection techniques. We then close by demonstrating an infinite family of congruences modulo 11 which is proven using a result of Ahlgren.

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arXiv2025

Convolutive sequences, I: Through the lens of integer partition functions

Motivated by the convolutive behavior of the counting function for partitions with designated summands in which all parts are odd, we consider coefficient sequences $(a_n)_{n\ge 0}$ of primitive eta-products that satisfy the generic convolutive property \begin{align*} \sum_{n\ge 0} a_{mn} q^n = \left(\sum_{n\ge 0} a_n q^n\right)^m \end{align*} for a specific positive integer $m$. Given the results of an exhaustive search of the Online Encyclopedia of Integer Sequences for such sequences for $m$ up to $6$, we first focus on the case where $m=2$ with our attention mainly paid to the combinatorics of two $2$-convolutive sequences, featuring bijective proofs for both. For other $2$-convolutive sequences discovered in the OEIS, we apply generating function manipulations to show their convolutivity. We also give two examples of $3$-convolutive sequences. Finally, we discuss other convolutive series that are not eta-products.

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arXiv2025

Elementary Proofs and Generalizations of Recent Congruences of Thejitha and Fathima

Motivated by recent work of Hirschhorn and the author, Thejitha and Fathima recently considered arithmetic properties satisfied by the function $a_5(n)$ which counts the number of integer partitions of weight $n$ wherein even parts come in only one color (i.e., they are monochromatic), while the odd parts may appear in one of five colors. They proved two sets of Ramanujan--like congruences satisfied by $a_5(n)$, relying heavily on modular forms. In this note, we prove their results via purely elementary means, utilizing generating function manipulations and elementary $q$-series dissections. We then extensively generalize these two sets of congruences to infinite families of divisibility properties in which the results of Thejitha and Fathima are specific instances.

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arXiv2025

Arithmetic properties of $t$-Schur overpartitions

In a recent work, Nadji and Ahmia introduced the $t$-Schur overpartitions as an overpartition analogue for $t$-Schur partitions, which generalizes the classical Schur's partitions into parts congruent to $1$ or $5$ modulo $6$. We continue the study of this new class of overpartitions and prove several arithmetic results for the cases $t=3,9$ and $t$ being a power of $2$ or a power of $3$.

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arXiv2025

Elementary Proofs of Two Congruences for Partitions with Odd Parts Repeated at Most Twice

In a recent article on overpartitions, Merca considered the auxiliary function $a(n)$ which counts the number of partitions of $n$ where odd parts are repeated at most twice (and there are no restrictions on the even parts). In the course of his work, Merca proved the following: For all $n\geq 0$, \begin{align*} a(4n+2) &\equiv 0 \pmod{2}, \textrm{\ \ and} \\ a(4n+3) &\equiv 0 \pmod{2}. \end{align*} Merca then indicates that a classical proof of these congruences would be very interesting. The goal of this short note is to fulfill Merca's request by providing two truly elementary (classical) proofs of these congruences.

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arXiv2025

Elementary Proofs of Recent Congruences for Overpartitions Wherein Non-Overlined Parts are Not Divisible by 6

We define $\overline{R_l^*}(n)$ as the number of overpartitions of $n$ in which non-overlined parts are not divisible by $l$. In a recent work, Nath, Saikia, and the second author established several families of congruences for $\overline{R_l^*}(n)$, with particular focus on the cases $l=6$ and $l=8$. In the concluding remarks of their paper, they conjectured that $\overline{R_6^*}(n)$ satisfies an infinite family of congruences modulo $128$. In this paper, we confirm their conjectures using elementary methods. Additionally, we provide elementary proofs of two congruences for $\overline{R_6^*}(n)$ previously proven via the machinery of modular forms by Alanazi, Munagi, and Saikia.

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arXiv2025

Extending Recent Congruence Results on $(\ell,μ)$-Regular Overpartitions

Recently, Alanazi, Munagi, and Saikia employed the theory of modular forms to investigate the arithmetic properties of the function $\overline{R_{\ell,μ}}(n)$, which enumerates the overpartitions of $n$ where no part is divisible by either $\ell$ or $μ$, for various integer pairs $(\ell, μ)$. In this paper, we substantially extend several of their results and establish infinitely many families of new congruences. Our proofs are entirely elementary, relying solely on classical $q$-series manipulations and dissection formulas.

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arXiv2025

Overpartitions and Kaur, Rana, and Eyyunni's mex sequences

Kaur, Rana, and Eyyunni recently defined the mex sequence of a partition and established, by analytic methods, connections to two disparate types of partition-related objects. We make a bijection between partitions with certain mex sequences and a uniform family of overpartitions which allows us to provide combinatorial proofs of their results, as they requested.

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arXiv2025

Arithmetic properties of MacMahon-type sums of divisors: the odd case

A century ago, P. A. MacMahon introduced two families of generating functions, $$ \sum_{1\leq n_1<n_2<\cdots<n_t}\prod_{k=1}^t\frac{q^{n_k}}{(1-q^{n_k})^2} \quad\text{ and } \sum_{\substack{1\leq n_1<n_2<\cdots<n_t\\ \text{$n_1,n_2,\dots,n_t$ odd}}}\prod_{k=1}^t\frac{q^{n_k}}{(1-q^{n_k})^2}, $$ which connect sum-of-divisors functions and integer partitions. These have recently drawn renewed attention. In particular, Amdeberhan, Andrews, and Tauraso extended the first family above by defining $$ U_t(a,q):=\sum_{1\leq n_1<n_2<\cdots<n_t}\prod_{k=1}^t\frac{q^{n_k}}{1+aq^{n_k}+q^{2n_k}} $$ for $a=0, \pm1, \pm2$ and investigated various properties, including some congruences satisfied by the coefficients of the power series representations for $U_t(a,q)$. These arithmetic aspects were subsequently expanded upon by the authors of the present work. Our goal here is to generalize the second family of generating functions, where the sums run over odd integers, and then apply similar techniques to show new infinite families of Ramanujan--like congruences for the associated power series coefficients.

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