Related papers: Notes for Neighborly Partitions

We use the celebrated circle method of Hardy and Ramanujan to develop convergent formulae for counting a restricted class of partitions that arise from the G\"ollnitz--Gordon identities.

Dyson's rank function and the Andrews--Garvan crank function famously give combinatorial witnesses for Ramanujan's partition function congruences modulo 5, 7, and 11. While these functions can be used to show that the corresponding sets of…

We obtain the generating functions for partial matchings avoiding neighbor alignments and for partial matchings avoiding neighbor alignments and left nestings. We show that there is a bijection between partial matchings avoiding three…

Recently, Rosengren utilized an integral method to prove a number of conjectural identities found by Kanade and Russell. Using this integral method, we give new proofs to some double sum identities of Rogers-Ramanujan type. These identities…

For a positive integer $r$, George Andrews proved that the set of partitions of $n$ in which odd multiplicities are at least $2r + 1$ is equinumerous with the set of partitions of $n$ in which odd parts are congruent to $2r + 1$ modulo $4r…

In a series of two papers, S. Capparelli, A. Meurman, A. Primc, M. Primc (CMPP) and then M. Primc put forth three remarkable sets of conjectures, stating that the generating functions of coloured integer partition in which the parts satisfy…

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…

In this paper we refine a weighted partition identity of Alladi. We write explicit formulas of generating functions for the number of partitions grouped with respect to a partition statistic other than the norm. We tie our weighted results…

We give $Z$-monomial generators for the vacuum spaces of certain level 2 standard modules of type $A^{(2)}_{\textrm{odd}}$ with indices running over integer partitions. In particular, we give a Lie theoretic interpretation of the…

We examine a method to conjecture two very famous identities that were conjectured by Ramanujan, and later found to be known to Rogers.

In this paper, we consider the set of partitions $ped(n)$ which counts the number of partitions of $n$ wherein the even parts are distinct (and the odd parts are unrestricted). Using an algorithm developed by Radu, we prove congruences…

We obtain congruences for the number a(n) of cubic partitions using modular forms. The notion of cubic partitions is introduced by Chan and named by Kim in connection with Ramanujan's cubic continued fractions. Chan has shown that a(n) has…

Neighborhood is an important concept in covering based rough sets. That under what condition neighborhoods form a partition is a meaningful issue induced by this concept. Many scholars have paid attention to this issue and presented some…

Recently, Amderberhan, Griffin, Ono, and Singh started the study of "traces of partition Eisenstein series" and used it to give explicit formulas for many interesting functions. In this note we determine the precise spaces in which they…

Adyan and Rabin showed that most properties of groups cannot be algorithmically recognized from a finite presentation alone. We prove that, if one is also given a solution to the word problem, then the class of fundamental groups of closed,…

In 1998, Michael Hirschhorn discovered 5-dissections of the Rogers--Ramanujan continued fraction $R(q)$ and its reciprocal. In this paper, we obtain the 5-dissections for functions $R(q)R(q^2)^2$ and $R(q)^2/R(q^2)$, which are essentially…

A partition statistic ` crank' gives combinatorial interpretations for Ramanujan's famous partition congruences. In this paper, we establish an asymptotic formula, Ramanujan type congruences, and q-series identities that the number of…

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 survey the techniques used in our recent resolution of the Kadison-Singer problem and proof of existence of Ramanujan Graphs of every degree: mixed characteristic polynomials and the method of interlacing families of polynomials. To…

The generating function of partitions with repeated (resp. distinct) parts such that each odd part is less than twice the smallest part is shown to be the third order mock theta function $\omega(q)$ (resp. $\nu(-q)$). Similar results for…