Related papers: M_2-rank differences for overpartitions

We develop the theory of copartitions, which are a generalization of partitions with connections to many classical topics in partition theory, including Rogers-Ramanujan partitions, theta functions, mock theta functions, partitions with…

Recently the author introduced two new integer partition quadruple functions, which satisfy Ramanujan-type congruences modulo $3$, $5$, $7$, and $13$. Here we reprove the congruences modulo $3$, $5$, and $7$ by defining a rank-type…

In 2003, Hammond and Lewis defined a statistic on partitions into 2 colors which combinatorially explains certain well known partition congruences mod 5. We give two analogs of Hammond and Lewis's birank statistic. One analog is in terms of…

We extend partition-theoretic work of Andrews, Bressoud, and Burge to overpartitions, defining the notions of successive ranks, generalized Durfee squares, and generalized lattice paths, and then relating these to overpartitions defined by…

New identities and congruences involving the ranks and cranks of partitions are proved. The proof depends on a new partial differential equation connecting their generating functions.

We show that the Zagier-Eisenstein series shares its non-holomorphic part with certain weak Maass forms whose holomorphic parts are generating functions for overpartition rank differences. This has a number of consequences, including exact…

In 2003, Atkin and Garvan initiated the study of rank and crank moments for ordinary partitions. These moments satisfy a strict inequality. We prove that a strict inequality also holds for the first rank and crank moments of overpartitions…

This paper has a two-fold purpose. First, by considering a reformulation of a deep theorem of G\"ollnitz, we obtain a new weighted partition identity involving the Rogers-Ramanujan partitions, namely, partitions into parts differing by at…

We study the Dyson rank function $N(r,3;n)$, the number of partitions of $n$ with rank $\equiv r \pmod 3$. We investigate the convexity of these functions. We extend $N(r,3;n)$ multiplicatively to the set of partitions, and we determine the…

We consider the symmetrized moments of three ranks and cranks, similar to the work of Garvan for the rank and crank of a partition. By using Bailey pairs and elementary rearrangements, we are able to find useful expressions for these…

Atkin and Garvan introduced the functions $N_k(n)$ and $M_k(n)$, which denote the $k$-th moments of ranks and cranks in the theory of partitions. Let $e_{2r}(n)$ be the $n$-th Fourier coefficient of $E_{2r}(\tau)/\eta(\tau)$, where…

In order to provide a unified combinatorial interpretation of congruences modulo $5$ for 2-colored partition functions, Garvan introduced a bicrank statistic in terms of weighted vector partitions. In this paper, we obtain some inequalities…

Bringmann and Lovejoy introduced a rank for overpartition pairs and investigated its role in congruence properties of $\bar{pp}(n)$, the number of overpartition pairs of n. In particular, they applied the theory of Klein forms to show that…

At the 1987 Ramanujan Centenary meeting Dyson asked for a coherent group-theoretical structure for Ramanujan's mock theta functions analogous to Hecke's theory of modular forms. Many of Ramanujan's mock theta functions can be written in…

Let $N(m,n)$ denote the number of partitions of $n$ with rank $m$, and let $M(m,n)$ denote the number of partitions of $n$ with crank $m$. Chan and Mao proved that for any nonnegative integers $m$ and $n$, $N(m,n)\geq N(m+2,n)$ and for any…

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…

In $1944$, Freeman Dyson defined the concept of rank of an integer partition and introduced without definition the term of crank of an integer partition. A definition for the crank satisfying the properties hypothesized for it by Dyson was…

We prove analytic and combinatorial identities reminiscent of Schur's classical partition theorem. Specifically, we show that certain families of overpartitions whose parts satisfy gap conditions are equinumerous with partitions whose parts…

In this paper we introduce k-run overpartitions as natural analogs to partitions without k-sequences, which were first defined and studied by Holroyd, Liggett, and Romik. Following their work as well as that of Andrews, we prove a number of…

In a recent work, Andrews defined the singular overpartitions with the goal of presenting an overpartition analogue to the theorems of Rogers--Ramanujan type for ordinary partitions with restricted successive ranks. As a small part of his…