Related papers: Some New Congruences For Overpartition Function Wi…

In this short note, we prove several new congruences for the overcubic partition triples function, using both elementary techniques and the theory of modular forms. These extend the recent list of such congruences given by Nayaka,…

Let $A_k(n)$ denote the set of $k$-distinct partitions of $n$, and let $B_k(n)$ be the set of $k$-regular partitions of $n$. Glaisher showed that $\# A_k(n) = \# B_k(n)$. For $k=2$, this equality yields the celebrated Euler's partition…

We present two new Ramanujan-type congruences modulo 5 for overpartition. We also give an affirmative answer to a conjecture of Dou and Lin, which includes four congruences modulo 25 for overpartition.

Suppose $\Pi$ is an exchangeable random partition of the positive integers and $\Pi_n$ is its restriction to $\{1, ..., n\}$. Let $K_n$ denote the number of blocks of $\Pi_n$, and let $K_{n,r}$ denote the number of blocks of $\Pi_n$…

It is well known that the Bell numbers represent the total number of partitions of an n-set. Similarly, the Stirling numbers of the second kind, represent the number of k-partitions of an n-set. In this paper we introduce a certain…

For nonzero coprime integers $a$ and $b$, a positive integer $\ell$ is said to be \emph{good with respect to $a$ and $b$} if there exists a positive integer $k$ such that $\ell$ divides $a^{k} + b^{k}$. The concept of good integers has been…

In this paper, we consider various theorems of P.A. MacMahon and M.V. Subbarao. For a non-negative integer $n$, MacMahon proved that the number of partitions of $n$ wherein parts have multiplicity greater than 1 is equal to the number of…

Let $k\geq\ell\geq1$ and $n\geq 1$ be integers. Let $G(k,n)$ be the complete $k$-partite graph with $n$ vertices in each colour class. An $\ell$-decomposition of $G(k,n)$ is a set $X$ of copies of $K_k$ in $G(k,n)$ such that each copy of…

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…

We study $\ell$-regular partitions by defining a sequence of modular forms of level $\ell$ and quadratic character which encode their $\ell$-adic behavior. We show that this sequence is congruent modulo increasing powers of $\ell$ to level…

In this paper we present a family of continued fraction expansions for $e^n$, with $n\ge 1$, with a simple expression having partial denominators given by arithmetic progressions. We give an estimate for the convergence speed showing that…

Noting a curious link between Andrews' even-odd crank and the Stanley rank, we adopt a combinatorial approach building on the map of conjugation and continue the study of integer partitions with parts separated by parity. Our motivation is…

A nonempty set $A\subset\mathbb{N}$ is $\ell$-strong Schreier if $\min A\geqslant \ell|A|-\ell+1$. We define a set of positive integers to be sparse if either the set has at most two numbers or the differences between consecutive numbers in…

The $(k,i)$-singular overpartitions, combinatorial objects introduced by Andrews in 2015, are known to satisfy Ramanujan-type congruences modulo any power of prime coprime to $6k$. In this paper we consider the parity of the number…

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…

Very recently, Thejitha, Sellers, and Fathima defined the function $a_{r,s}(n)$, which enumerates the number of multicolored partitions of $n$, wherein both even parts and odd parts may appear in one of $r$-colors and $s$-colors,…

Let $\mathrm{pod}_{-4}(n)$ denote the number of partition quadruples of $n$ where the odd parts in each partition are distinct. We find many arithmetic properties of $\mathrm{pod}_{-4}(n)$ involving the following infinite family of…

In this paper, we obtain inequalities on $M_2$-ranks of overpartitions modulo $6$. Let $\overline{N}_2(s,m,n)$ to be the number of overpartitions of $n$ whose $M_2$-rank is congruent to $s$ modulo $m$. For $M_2$-ranks modulo $3$, Lovejoy…

Let $\mathcal{A}$ and $\matchcal{B}$ denote two families of subsets of an $n$-element set. The pair $(\mathcal{A},\mathcal{B})$ is said to be $\ell$-cross-intersecting iff $|A\cap B| = \ell$ for all $A\in\mathcal{A}$ and $B\in\mathcal{B}$.…

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 $a_3(n)$. Recently, both authors…