Related papers: Some crystal Rogers-Ramanujan type identities

The partition statistic $V_R$-rank is introduced to give combinatorial proofs of the Ramanujan type congruences mod 3 for certain classes of partition functions.

In this paper we give a combinatorial parametrization of leading terms of defining relations for level $k$ standard modules for affine Lie algebra of type $C_{n}\sp{(1)}$. Using this parametrization we conjecture colored Rogers-Ramanujan…

Regular $A_n$-crystals are certain edge-colored directed graphs which are related to representations of the quantized universal enveloping algebra $U_q(\mathfrak{sl}_{n+1})$. For such a crystal $K$ with colors $1,2,...,n$, we consider its…

We prove four new Rogers-Ramanujan-type identities for double series. They follow from the classical Rogers-Ramanujan identities using the constant term method and properties of Rogers-Szeg\H{o} polynomials.

George Andrews [\emph{Bull. Amer. Math. Soc.}, 2007, 561--573] introduced the idea of a \emph{signed partiton} of an integer; similar to an ordinary integer partitions, but where some of the parts could be negative. Further, Andrews…

We provide the explicit combinatorial structure of the Kirillov-Reshetikhin crystals B^{r,s} of type D_n(1), B_n(1), and A_{2n-1}(2). This is achieved by constructing the crystal analogue sigma of the automorphism of the D_n(1) (resp.…

Lusztig's theory of PBW bases gives a way to realize the infinity crystal for any simple complex Lie algebra where the underlying set consists of Kostant partitions. In fact, there are many different such realizations, one for each reduced…

Partitions with distinct even parts have long been the subject of extensive research. In this paper, We present some new perspectives on such partitions from a combinatorial viewpoint, and connect them with signed partitions and bicolored…

For any positive integers $n$ and $r$, let $p_r(n)$ denotes the number of partitions of $n$ where each part has $r$ distinct colours. Many authors studied the partition function $p_r(n)$ for particular values of $r$. In this paper, we prove…

We prove a number of new Rogers-Ramanujan type identities involving double, triple and quadruple sums. They were discovered after an extensive search using Maple. The main idea of proofs is to reduce them to some known identities in the…

We use a q-series identity by Ramanujan to give a combinatorial interpretation of Ramanujan's tau function which involves t-cores and a new class of partitions which we call (m,k)-capsids. The same method can be applied in conjunction with…

This is the first of a series of papers studying combinatorial (with no ``subtractions'') bases and characters of standard modules for affine Lie algebras, as well as various subspaces and ``coset spaces'' of these modules. In part I we…

A cubic partition is an integer partition wherein the even parts can appear in two colors. In this paper, we introduce the notion of generalized cubic partitions and prove a number of new congruences akin to the classical Ramanujan-type. We…

Via the contour integral method, we establish a reduction formula from a double series to a single series with parameters, which not only implies Uncu and Zudilin's two results and Cao and Wang's two results, but also is related to…

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…

Recently, Andrews proved two conjectures on a partition statistic introduced by Beck. Very recently, Chern established some results on weighted rank and crank moments and proved many Andrews-Beck type congruences. Motivated by Andrews and…

The Ram-Yip formula for Macdonald polynomials (at t=0) provides a statistic which we call charge. In types A and C it can be defined on tensor products of Kashiwara-Nakashima single column crystals. In this paper we prove that the charge is…

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…

In this article, we study the arithmetic properties of the partition function $p_8(n)$, the number of 8-colour partitions of $n$. We prove several Ramanujan type congruences modulo higher powers of 2 for the function $p_8(n)$ by finding…

We establish some new bilateral double-sum Rogers-Ramanujan identities involving parameters. As applications, these identities yield several new multi-sum Rogers-Ramanujan type identities. Our proofs utilize the theory of basic…