Related papers: Some Congruences of a Restricted Bipartition Funct…

Amdeberhan et al. (2024) introduced the notion of a generalized overcubic partition function $\overline a_c (n)$ and proved an infinite family of congruences modulo a prime $p\ge 3$ and some Ramanujan type congruences. In this paper, we…

A proof of several identities of Ramanujan involving theta functions of level $7$ is given which uses a specific modular function for $\Gamma_1(7)$ and Klein's projective representation of $PSL(2,7)$ into $PSL(3, \mathbb{C})$. Four…

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}, \\…

We consider the Lommel functions $s_{\mu,\nu}(z)$ for different values of the parameters $(\mu,\nu)$. We show that if $(\mu,\nu)$ are half integers, then it is possible to describe these functions with an explicit combination of polynomials…

Let $ B_{\ell}(n)$ denote the number of $\ell$-regular bipartitions of $n.$ In this article, we prove that $ B_{\ell}(n)$ is always almost divisible by $p_i^j$ if $p_i^{2a_i}\geq \ell,$ where $j$ is a fixed positive integer and…

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…

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…

It is well known that Ramanujan conjectured congruences modulo powers of $5$, $7$ and and $11$ for the partition function. These were subsequently proved by Watson (1938) and Atkin (1967). In 2009 Choi, Kang, and Lovejoy proved congruences…

Recently Lachterman, Schayer, and Younger published an elegant proof of the Ramanujan congruences for the partition function $p(n)$. Their proof uses only the classical theory of modular forms as well as a beautiful result of Choie, Kohnen,…

Recently Amdeberhan, Sellers, and Singh introduced a new infinite family of partition functions called generalized cubic partitions. Given a positive integer $d$, they let $a_d(n)$ be the counting function for partitions of $n$ in which the…

For any relatively prime integers $r$ and $s$, let $a_{r,s}(n)$ denote the number of $(r,s)$-regular partitions of a positive integer of $n$ into distinct parts. Prasad and Prasad (2018) proved many infinite families of congruences modulo 2…

Andrews and the third author recently studied congruences for certain restricted two-color partitions. They made two conjectures for Ramanujan-type congruences and a vanishing identity for the limiting sequence. In this paper, we settle…

Let ${{\overline{p}}_{3}}(n)$ be the number of overpartition triples of $n$. By elementary series manipulations, we establish some congruences for ${\overline{p}}_{3}(n)$ modulo small powers of 2, such as…

Ramanujan's congruence $p(5k+4) \equiv 0 \pmod 5$ led Dyson \cite{dyson} to conjecture the existence of a measure "rank" such that $p(5k+4)$ partitions of $5k+4$ could be divided into sub-classes with equal cardinality to give a direct…

The study of Ramanujan-type congruences for functions specific to additive number theory has a long and rich history. Motivated by recent connections between divisor sums and overpartitions via congruences in arithmetic progressions, we…

Fix a partition $\mu=(\mu_1,\dotsc,\mu_m)$ of an integer $k$ and positive integer $d$. For each $n>k$, let $\chi^\lambda_\mu$ denote the value of the irreducible character of $S_n$ at a permutation with cycle type…

Let v(n) denote the number of compositions (ordered partitions) of a positive integer n into powers of 2. It appears that the function v(n) satisfies many congruences modulo 2^N. For example, for every integer B there exists (as k tends to…

In his 1984 AMS Memoir, George Andrews defined the family of $k$--colored generalized Frobenius partition functions. These are denoted by $c\phi_k(n)$ where $k\geq 1$ is the number of colors in question. In that Memoir, Andrews proved…

For a positive integer $n$, let $B_{\ell_1,\dots,\ell_r}(n)$ denote the number of $(\ell_1,\ell_2,\cdots,\ell_r)$-regular multipartitions of $n$. If $\ell_1=\ell_2=\cdots=\ell_r=\ell$, then we denote $B_{\ell_1,\dots,\ell_r}(n)$ as…

Let $p_k(n)$ denote the number of $2$-color partitions of $n$ where one of the colors appears only in parts that are multiples of $k$. We will prove a conjecture of Ahmed, Baruah, and Dastidar on congruences modulo $5$ for $p_k(n)$.…