Related papers: Distinct parts partitions without sequences

We study the generating function of the excess number of Rogers-Ramanujan partitions with odd rank over those with even rank, and, using combinatorial and analytical techniques, show that this generating function is closely connected with…

Strict partitions are enumerated with respect to the weight, the number of parts, and the number of sequences of odd length. We write this trivariate generating function as a double sum $q$-series. Equipped with such a combinatorial set-up,…

Recently, Andrews introduced separable integer partition classes and studied some well-known theorems. In this article, we will consider the types of partitions with restrictions on consecutive parts. We will show that such partitions are…

We set up a combinatorial framework for inclusion-exclusion on the partitions into distinct parts to obtain an alternative generating function of partitions into distinct and non-consecutive parts. In connection with Rogers-Ramanujan…

The number of parts in the partitions (resp. distinct partitions) of $n$ with parts from a set were considered. Its generating functions were obtained. Consequently, we derive several recurrence identities for the following functions: the…

Andrews studied a function which appears in Ramanujan's identities. In Ramanujan's "Lost" Notebook, there are several formulas involving this function, but they are not as simple as the identities with other similar shape of functions.…

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 focus on writing closed forms of generating functions for the number of partitions with gap conditions as double sums starting from a combinatorial construction. Some examples of the sets of partitions with gap conditions to be discussed…

In 2017, Keith presented a comprehensive survey on integer partitions into parts that are simultaneously regular, distinct, and/or flat. Recently, the authors initiated a study of partitions into parts that are simultaneously regular and…

We study enumeration functions for unimodal sequences of positive integers, where the size of a sequence is the sum of its terms. We survey known results for a number of natural variants of unimodal sequences, including Auluck's generalized…

Motivated by the study of integer partitions, we consider partitions of integers into fractions of a particular form, namely with constant denominators and distinct odd or even numerators. When numerators are odd, the numbers of partitions…

The celebrated Rogers-Ramanujan identities equate the number of integer partitions of $n$ ($n\in\mathbb N_0$) with parts congruent to $\pm 1 \pmod{5}$ (respectively $\pm 2 \pmod{5}$) and the number of partitions of $n$ with super-distinct…

In this paper, we introduce the generating functions of partition sequences. Partition sequences have a one-to-one correspondence with partitions. Therefore, the generating function has no multiplicity and appears meaningless initially.…

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…

We prove formulas for the generating functions for M_2-rank differences for partitions without repeated odd parts. These formulas are in terms of modular forms and generalized Lambert series.

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,…

In the first three papers, we conducted a series of discussions on the statistics of strict partitions and Rogers-Ramanujan partitions, specifically the sequences of odd length (denoted as $\mathrm{sol}$) and its extensions. We established…

Basis partitions are minimal partitions corresponding to successive rank vectors. We show combinatorially how basis partitions can be generated from primary partitions which are equivalent to the Rogers-Ramanujan partitions. This leads to…

For any positive integers $s$ and $t$, let $Q_{t}^{s}(n)$ denotes the number of partitions of a positive integer $n$ into distinct parts such that no part is congruent to $s$ or $t-s$ modulo $t$. We prove some Ramanujan-type congruences for…

We give a series of recursive identities for the number of partitions with exactly $k$ parts and with constraints on both the minimal difference among the parts and the minimal part. Using these results we demonstrate that the number of…