Related papers: Separable integer partition classes and partitions…

Let $m\ge 2$ be a fixed positive integer. Suppose that $m^j \leq n< m^{j+1}$ is a positive integer for some $j\ge 0$. Denote $b_{m}(n)$ the number of $m$-ary partitions of $n$, where each part of the partition is a power of $m$. In this…

We investigate the number $N_{d,r}(s)$ of $(s, s+r)$-core integer partitions with $d$-distinct parts. Our first main result is a proof of a recurrence relation conjectured by Sahin in 2018. We also derive generating functions, asymptotics,…

Using elementary methods, we prove new formulas for $\operatorname{pp}(n)$, the number of plane partitions of $n$, $\operatorname{pp}_r(n)$, the number of plane partitions of $n$ with at most $r$ rows, $\operatorname{pp}^s(n)$, the number…

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 1969, Andrews proved a theorem on partitions with difference conditions which generalises Schur's celebrated partition identity. In this paper, we generalise Andrews' theorem to overpartitions. The proof uses q-differential equations and…

Given a prime number $p$, the study of divisibility properties of a sequence $c(n)$ has two contending approaches: $p$-adic valuations and superconcongruences. The former searches for the highest power of $p$ dividing $c(n)$, for each $n$;…

For rational $\alpha$, the fractional partition functions $p_\alpha(n)$ are given by the coefficients of the generating function $(q;q)^\alpha_\infty$. When $\alpha=-1$, one obtains the usual partition function. Congruences of the form…

In this paper we explore Kruyswijk's method and show how to obtain congruences for cubic partition. That apart we also examine inequalities for a(n) and provide upper bound for it in the fashion of the classic partition function p(n).

We introduce the multiset partition algebra, ${\rm M\!P}_{r,k}(x)$, that has bases elements indexed by multiset partitions, where $x$ is an indeterminate and $r$ and $k$ are non-negative integers. This algebra can be realized as a diagram…

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…

An $(n,k)$-Sperner partition system is a set of partitions of some $n$-set such that each partition has $k$ nonempty parts and no part in any partition is a subset of a part in a different partition. The maximum number of partitions in an…

This paper provides algebraic proofs for several types of congruences involving the multipartition function and self-convolutions of the divisor function. Our computations use methods of Differential Algebra in $\mathbb{Z}/q\mathbb{Z}$,…

In this paper, we show that the difference between the number of parts in the odd partitions of $n$ and the number of parts in the distinct partitions of $n$ satisfies Euler's recurrence relation for the partition function $p(n)$ when $n$…

In this work, we investigate the arithmetic properties of $p_{1,7^k}(n)$, which counts 2-color partitions of $n$ where one of the colors appears only in parts that are multiples of $7^k$. By constructing generating functions for…

For positive integers $k, l \geq 2$, the set of $k$-regular partitions in which parts appear at most $l$ times has attracted a lot of interest in that a composition of Glaisher's mapping can be used to prove the associated partition…

The number of partitions of n into parts divisible by a or b equals the number of partitions of n in which each part and each difference of two parts is expressible as a non-negative integer combination of a or b. This generalizes…

Andrews and El Bachraoui recently studied various two-colored integer partitions, including those related to two-colored partitions into distinct parts with constraints and overpartitions. Their work raised questions about the existence of…

For all positive integers $k,l,n$, the Little Glaisher theorem states that the number of partitions of $n$ into parts not divisible by $k$ and occurring less than $l$ times is equal to the number of partitions of $n$ into parts not…

We prove several Ramanujan-type congruences modulo powers of $5$ for partition $k$-tuples with $5$-cores, for $k=2, 3, 4$. We also prove some new infinite families of congruences modulo powers of primes for $k$-tuples with $p$-cores, where…

As a well-known enumerative problem, the number of solutions of the equation $m=m_1+...+m_k$ with $m_1\leqslant...\leqslant m_k$ in positive integers is $\Pi(m,k)=\sum_{i=0}^k\Pi(m-k,i)$ and $\Pi$ is called the additive partition function.…