Related papers: Congruences for the partition function $\text{PDO}…

In 2007, Andrews and Paule introduced the family of functions $\Delta_k(n)$, which enumerate the number of broken $k$-diamond partitions for a fixed positive integer $k$. In 2013, Radu and Sellers completely characterized the parity of…

Let $p_{k}(n)$ be the coefficient of $q^n$ in the series expansion of $(q;q)_{\infty}^{k}$. It is known that the partition function $p(n)$, which corresponds to the case when $k=-1$, satisfies congruences such as $p(5n+4)\equiv 0\pmod{5}$.…

The coefficients of the generating function $(q;q)^\alpha_\infty$ produce $p_\alpha(n)$ for $\alpha \in \mathbb{Q}$. In particular, when $\alpha = -1$, the partition function is obtained. Recently, Chan and Wang identified and proved…

Recently, Andrews, Hirschhorn and Sellers have proven congruences modulo 3 for four types of partitions using elementary series manipulations. In this paper, we generalize their congruences using arithmetic properties of certain quadratic…

Let $p(n)$ be the ordinary partition function. In the 1960s Atkin found a number of examples of congruences of the form $p( Q^3 \ell n+\beta)\equiv0\pmod\ell$ where $\ell$ and $Q$ are prime and $5\leq \ell\leq 31$; these lie in two natural…

Let p(n, k) denote the number of partitions of n into parts less than or equal to k. We show several properties of this function modulo 2. First, we prove that for fixed positive integers k and m, p(n,k) is periodic modulo m. Using this, we…

For integers $0 < r \leq t$, let the function $D_{r,t}(n)$ denote the number of parts among all partitions of $n$ into distinct parts that are congruent to $r$ modulo $t$. We prove the asymptotic formula $$D_{r,t}(n) \sim \dfrac{3^{\frac…

We obtain congruences for the number a(n) of cubic partitions using modular forms. The notion of cubic partitions is introduced by Chan and named by Kim in connection with Ramanujan's cubic continued fractions. Chan has shown that a(n) has…

Numerous congruences for partitions with designated summands have been proven since first being introduced and studied by Andrews, Lewis, and Lovejoy. This paper explicitly characterizes the number of partitions with designated summands…

Let $pod_2(n)$ denote the number of $2$-regular partitions of $n$ with distinct odd parts (even parts are unrestricted). In this article, we obtain congruences for $pod_2(n)$ mod $2$ and mod $8$ using some generating function manipulations…

We denote the number of partitions of $n$ wherein the even parts are distinct (and the odd parts are unrestricted) by $ped(n)$. In this paper, we will use generating function manipulations to obtain new congruences for $ped(n)$ modulo $24$.

Let $a_3(n)$ and $a_9(n)$ are 3 and 9-regular cubic partitions of $n$. In this paper, we find the infinite family of congruences modulo powers of 3 for $a_3(n)$ and $a_9(n)$ such as \[a_3\left (3^{2\alpha}n+\frac{3^{2\alpha}-1}{4}\right…

In 1967, Atkin and O'Brien conjectured congruences for the partition function involving Hecke operators modulo powers of 13. In this paper, we provide a simple proof of this conjecture.

In 2017, Andrews, Dixit, Schultz and Yee introduced the function $\overline{\textrm{spt}}_\omega(n)$, which denotes the number of smallest parts in the overpartitions of $n$ in which the smallest part is always overlined and all odd parts…

Let $p(n)$ denote the partition function. In this article, we will show that congruences of the form $$ p(m^j\ell^kn+B)\equiv 0\mod m \text{for all} n\ge 0 $$ exist for all primes $m$ and $\ell$ satisfying $m\ge 13$ and $\ell\neq 2,3,m$.…

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 $p_k(n)$.…

A partition of $n$ is called a $t$-core partition if none of its hook number is divisible by $t.$ In 2019, Hirschhorn and Sellers \cite{Hirs2019} obtained a parity result for $3$-core partition function $a_3(n)$. Motivated by this result,…

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…

Let $\overline{p}_o(n)$ denote the number of overpartitions of $n$ into odd parts. The partition function $\overline{p}_o(n)$ has been the subject of many recent studies where many explicit Ramanujan-like congruences were discovered. In…

Stanley defined a partition function t(n) as the number of partitions $\lambda$ of n such that the number of odd parts of $\lambda$ is congruent to the number of odd parts of the conjugate partition $\lambda'$ modulo 4. We show that t(n)…