Related papers: Two identities involving the cubic partition funct…
Let $b(n)$ denote the number of cubic partition pairs of $n$. In this paper, we aim to provide a strategy to obtain arithmetic properties of $b(n)$. This gives affirmative answers to two of Lin's conjectures.
We introduce two new integer partition functions, both of which are the number of partition quadruples of $n$ with certain size restrictions. We prove both functions satisfy Ramanujan-type congruences modulo $3$, $5$, $7$, and $13$ by use…
Recently, Wang and Ma propose a conjecture associated with the possible generalization of Andrews-Warnaar identities. It is confirmed in this paper. As the applications of this conjecture, we prove that a family of series can be expressed…
The cubic partitions of a natural number $n$, introduced by Chan and Kim, have generating function $\sum_{n=0}^{\infty}a(n)q^n= \frac{1}{(q; q)_{\infty}(q^2; q^2)_{\infty}}.$ In this paper, we generalize some results of Chen-Lin, which…
In this article we will derive a combinatorial formula for the partition function p(n). In the second part of the paper we will establish connection between partitions and q-binomial coefficients and give new interpretation for q-binomial…
By applying the derivative operators to Chu-Vandermonde convolution, several general harmonic number identities are established.
We prove a theorem which add a new member to Rogers-Ramanujan identities. This new member counts partitions with different type of constraints on even and odd parts. Generalizing this theorem, we obtain two family of partition identities of…
Given two polynomials f(x) and g(x), we extend the formula expressing the remainder in terms of the roots of these two polynomials to the case where f(x) is a Laurent polynomial. This allows us to give new expressions of a Schur function,…
A recent paper of Bump, McNamara and Nakasuji introduced a factorial version of Tokuyama's identity, expressing the partition function of a six vertex model as the product of a t-deformed Vandermonde and a Schur function. Here we provide an…
In this work, we construct a new Bailey pairs for the integral pentagon identity in terms of q-hypergeometric functions. The pentagon identity considered here represents equality of the partition functions of a certain three-dimensional…
A conceptual framework involving partition functions of normal factor graphs is introduced, paralleling a similar recent development by Al-Bashabsheh and Mao. The partition functions of dual normal factor graphs are shown to be a Fourier…
Recently, Merca and Yee proved some partition identities involving two new partition statistics. We refine these statistics and generalize the results of Merca and Yee. We also correct a small mistake in a result of Merca and Yee.
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…
We study the generating functions for cylindric partitions with profile $(c_1,c_2,c_3)$ for all $c_1,c_2,c_3$ such that $c_1+c_2+c_3=5$. This allows us to discover and prove seven new $A_2$ Rogers-Ramanujan identities modulo $8$ with…
In a recent paper by the authors, a bounded version of Goellnitz's (big) partition theorem was established. Here we show among other things how this theorem leads to nontrivial new polynomial analogues of certain fundamental identities of…
We give a commutative algebra viewpoint on Andrews recursive formula for the partitions appearing in "Gordon's identities", which are a generalization of Rogers-Ramanujan identities. Using this approach and differential ideals we conjecture…
Alladi studied partition theoretic implications of a two variable generalization of the Lebesgue identity. In this short note, we focus on a slight variation of the basic hypergeometric sum that Alladi studied. We present two new partition…
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,…
We prove polynomial boson-fermion identities for the generating function of the number of partitions of $n$ of the form $n=\sum_{j=1}^{L-1} j f_j$, with $f_1\leq i-1$, $f_{L-1} \leq i'-1$ and $f_j+f_{j+1}\leq k$. The bosonic side of the…
Using a probabilistic approach, we derive several interesting identities involving beta functions. Our results generalize certain well-known combinatorial identities involving binomial coefficients and gamma functions.