Related papers: Separable overpartition classes and excludant size…
Recently, Andrews and Newman studied the minimal excludant of a partition, which is defined as the smallest positive integer that is not a part of a partition. In this article, we consider the minimal excludant size of an overpartition,…
A partition of a positive integer $n$ is a non-increasing sequence of positive integers which sum to $n$. A recently studied aspect of partitions is the minimal excludant of a partition, which is defined to be the smallest positive integer…
Inspired by Andrews' and Bachraoui's work on partitions with repeated smallest part, we extend the concept to overpartitions. We study overpartitions with the restriction that the smallest non-overlined part appears exactly $k$ times and…
Recently, Andrews considered the partitions with parts separated by parity, in which parts of a given parity are all smaller than those of the other parity. Inspired from the partitions with parts separated by parity, we investigate the…
In this article, we first investigate the partitions whose parts are congruent to $a$ or $b$ modulo $k$ with the aid of separable integer partition classes with modulus $k$ introduced by Andrews. Then, we introduce the…
This paper is devoted to the study of sequences in overpartitions and their relation to 2-color partitions. An extensive study of a general class of double series is required to achieve these ends.
In his classic text, \emph{Combinatory Analysis}, MacMahon defined a perfect partition of a positive integer $n$ as a partition whose parts contain exactly one partition of every positive integer not exceeding $n$. In this paper we apply…
Recently, Chen, He, Hu and Xie considered the parity of the number of non-overlined (resp. overlined) parts of size greater than or equal to the size of the smallest overlined (resp. non-overlined) part in an overpartition. In this article,…
Inspired by Andrews' and Newman's work on the minimal excludant or "mex" of partitions, we define four new classes of minimal excludants for overpartitions and establish relations to certain functions due to Ramanujan.
In this paper, we focus on families of bipartitions, i.e. set partitions consisting of at most two components. We say that a family of bipartitions is a separating family for a set $S$ if every two elements in $S$ can be separated by some…
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…
Let $\overline{\mathrm{spt}}k(n)$ denote the number of overpartitions of $n$ where the smallest non-overlined part, say $s(\pi)$, appears $k$ times and every overlined part is bigger than $s(\pi)$. Let $\overline{\mathrm{spt}}k_o(n)$ denote…
We study the combinatorics of two classes of basic hypergeometric series. We first show that these series are the generating functions for certain overpartition pairs defined by frequency conditions on the parts. We then show that when…
Singular overpartitions, which are defined by George Andrews, are overpartitions whose Frobenius symbols have at most one overlined entry in each row. In his paper, Andrews obtained interesting combinatorial results on singular…
Kaur, Rana, and Eyyunni recently defined the mex sequence of a partition and established, by analytic methods, connections to two disparate types of partition-related objects. We make a bijection between partitions with certain mex…
Define the minimal excludant of an overpartition $\pi$, denoted $ \overline{\text{mex}}(\pi)$, to be the smallest positive integer that is not a part of the non-overlined parts of $\pi$. For a positive integer $n$, the function…
In this paper, we study the partition functions $\overline{R_\ell^\ast}(n)$, which count the number of overpartitions of $n$ where the non-overlined parts are $\ell$-regular for a given $\ell$. Using elementary techniques, as well as the…
Recently, Andrews introduced separable integer partition classes and analyzed some well-known theorems. In this paper, we investigate partitions with parts separated by parity introduced by Andrews with the aid of separable integer…
We introduce and study block-separated overpartitions, a constrained family of overpartitions in which no two consecutive distinct part-blocks are both overlined. This local restriction produces a new sequence that naturally interpolates…
In this paper, we generalize Andrews' partitions separated by parity to overpartitions in two ways. We investigate the generating functions for 16 overpartition families whose parts are separated by parity, and we prove various $q$-series…