Related papers: Partition identities associated to Rogers-Ramanuja…
We study a family of inhomogeneous Ising chain models along with an equivalent family of nearest neighbour particle systems. By the correspondence between the two families we prove identities of combinatorial significance relating to…
We study certain arithmetic properties of an analogue $B(n)$ of Lin's restricted partition function that counts the number of partition triples $\pi=(\pi_1,\pi_2,\pi_3)$ of $n$ such that $\pi_1$ and $\pi_2$ comprise distinct odd parts and…
In this note, we present a curious $q$-series identity with applications to certain partitions with bounded part differences.
The Rogers-Ramanujan identities are investigated using the Cauchy identity for Schur functions.
We derive two general transformations for certain basic hypergeometric series from the recurrence formulae for the partial numerators and denominators of two $q$-continued fractions previously investigated by the authors. By then…
In a recent work, Andrews gave analytic proofs of two conjectures concerning some variations of two combinatorial identities between partitions of a positive integer into odd parts and partitions into distinct parts discovered by Beck.…
Relations involving the Rogers-Ramanujan continued fractions $R(q),$ $R(q^3 ),$ and $R(q^4)$ are used to find new generating functions and congruences modulo 5 and 25 for 3-core, 4-core, 4-regular, and colored partition functions.
A q-series with nonnegative power series coefficients is called positive. The partition statistics BG-rank is defined as an alternating sum of parities of parts of a partition. It is known that the generating function for the number of…
We provide a general and unified combinatorial framework for a number of colored partition identities, which include the five, recently proved analytically by B. Berndt, that correspond to the exceptional modular equations of prime degree…
Finding an Andrews--Gordon type generating function identity for a linked partition ideal is difficult in most cases. In this paper, we will handle this problem in the setting of graph theory. With the generating function of directed graphs…
We prove seven of the Rogers-Ramanujan type identities modulo $12$ that were conjectured by Kanade and Russell. Included among these seven are the two original modulo $12$ identities, in which the products have asymmetric congruence…
Using sequences of finite length with positive integer elements and the inversion statistic on such sequences, a collection of binomial and multinomial identities are extended to their $q$-analog form via combinatorial proofs. Using the…
Partition identities are often statements asserting that the set $\mathcal P_X$ of partitions of $n$ subject to condition $X$ is equinumerous to the set $\mathcal P_Y$ of partitions of $n$ subject to condition $Y$. A Beck-type identity is a…
We prove an identity about partitions involving new combinatorial coefficients. The proof given is using a generating function. As an application we obtain the explicit expression of two shifted symmetric functions, related with Jack…
A proof of the first Rogers-Ramanujan identity is given using admissible neighborly partitions. This completes a program initiated by Mohsen and Mourtada. The admissible neighborly partitions involve an unusual mod 3 condition on the parts.
In this paper we study $b_5(n)$, the $5$-regular partitions of $n$. Using the theory of modular forms, we prove several theorems on the divisibility and distribution properties of $b_5(n)$ modulo prime $m\geq5$. In particular, we prove that…
We define a class of expressions for the multiple zeta function, and show how to determine whether an expression in the class vanishes identically. The class of such identities, which we call partition identities, is shown to coincide with…
Euler's identity equates the number of partitions of any non-negative integer n into odd parts and the number of partitions of n into distinct parts. Beck conjectured and Andrews proved the following companion to Euler's identity: the…
In this paper, we prove a new Rogers-Ramanujan-type identity, involving grounded partitions, by computing a character of the affine Kac-Moody algebra $D_4^{(3)}$ in two different ways. The product side is derived using Lepowsky's product…
We conjecture polynomial identities which imply Rogers--Ramanujan type identities for branching functions associated with the cosets $({\cal G}^{(1)})_{\ell-1}\otimes ({\cal G}^{(1)})_{1} / ({\cal G}^{(1)})_{\ell}$, with ${\cal…