Related papers: $6$-regular partitions: new combinatorial properti…

The main result of the paper is the existence of an infinitely many families of Ramanujan-type congruences for $b_4(n)$ and $b_6(n)$ modulo primes $m \geq 2$ and $m \geq 5$, respectively. We provide new examples of congruences for $b_4(n)$…

It is proved that the number of 9-regular partitions of n is divisible by 3 when n is congruent to 3 mod 4, and by 6 when n is congruent to 13 mod 16. An infinite family of congruences mod 3 holds in other progressions modulo powers of 4…

A partition of $n$ is $l$-regular if none of its parts is divisible by $l$. Let $b_l(n)$ denote the number of $l$-regular partitions of $n$. In this paper, using theta function identities due to Ramanujan, we establish some new infinite…

Let $b_{\ell, k}(n), b_{\ell, k, r}(n)$ count the number of $(\ell, k)$, $(\ell, k, r)$-regular partitions respectively. In this paper we shall derive infinite families of congruences for $b_{\ell, k}(n)$ modulo $2$ when $ (\ell, k) =…

For a positive integer $t\geq 2$, let $b_{t}(n)$ denote the number of $t$-regular partitions of a nonnegative integer $n$. Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo $2$…

Linear inequalities involving Euler's partition function $p(n)$ have been the subject of recent studies. In this article, we consider the partition function $Q(n)$ counting the partitions of $n$ into distinct parts. Using truncated theta…

The partition function $pod(n)$ enumerates the partitions of $n$ wherein odd parts are distinct and even parts are unrestricted. Recently, a number of properties for $pod(n)$ have been established. In this paper, for $k\in\{0,2\}$ we…

Let $ped(n)$ denote the number of partitions of $n$ wherein even parts are distinct (and odd parts are unrestricted). We show infinite families of congruences for $ped(n)$ modulo $8$. We also examine the behavior of $ped_{-2}(n)$ modulo $8$…

Let $b_3(n)$ be the number of $3$-regular partitions of $n$. Recently, W. J. Keith and F. Zanello discovered infinite families of Ramanujan type congruences modulo $2$ for $b_3(2n)$ involving every prime $p$ with $p \equiv 13, 17, 19, 23…

Let $p_k(n)$ be given by the $k$-th power of the Euler Product $\prod _{n=1}^{\infty}(1-q^n)^k=\sum_{n=0}^{\infty}p_k(n)q^{n}$. By investigating the properties of the modular equations of the second and the third order under the Atkin…

Recently, Hirschhorn and Sellers defined the partition function $a_r(n)$, which counts the number of partitions of $n$ wherein even parts come in only one color, while the odd parts may appear in one of $r$-colors for fixed $r\ge1$. The aim…

For a positive integer $\ell$, let $b_{\ell}(n)$ denote the number of $\ell$-regular partitions of a nonnegative integer $n$. Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo…

Let $b_k(n)$ denote the $k-$regular partitons of a natural number $n$. In this paper, we study the behavior of $b_k(n)$ modulo composite integers $M$ which are coprime to $6$. Specially, we prove that for arbitrary $k-$regular partiton…

Let $b_\ell(n)$ be the number of $\ell$-regular partitions of $n$. Recently, Hou et al established several infinite families of congruences for $b_\ell(n)$ modulo $m$, where $(\ell,m)=(3,3),(6,3),(5,5),(10,5)$ and $(7,7)$. In this paper, by…

We study certain arithmetic properties of an analogue $B(n)$ of Lin's restricted partition function that counts the number of partition triples $\pi=(\pi_1,\pi_2,\pi_3)$ of $n$ such that $\pi_1$ and $\pi_2$ comprise distinct odd parts and…

Recently, Lin introduced two new partition functions PD$_t(n)$ and PDO$_t(n)$, which count the total number of tagged parts over all partitions of $n$ with designated summands and the total number of tagged parts over all partitions of $n$…

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…

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,…

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…

In this paper we study the function $b_3(n)$ and $b_5(n)$, which denote the number of $3$-regular partitions and $5$-regular partitions of $n$ respectively. Using the theory of modular forms, we prove several arithmetic properties of…