Related papers: Parity Considerations in Rogers-Ramanujan-Gordon T…
We define an overpartition analogue of Gaussian polynomials (also known as $q$-binomial coefficients) as a generating function for the number of overpartitions fitting inside the $M \times N$ rectangle. We call these new polynomials over…
Let $p_{o}(n)$ (resp. $p_{e}(n)$) denote the number of partitions of $n$ with more odd parts (resp. even parts) than even parts (resp. odd parts). Recently, Kim, Kim, and Lovejoy proved that $p_{o}(n)>p_{e}(n)$ for all $n>2$ and conjectured…
Motivated by spin modular representations of the symmetric groups, we propose two generalizations of the Schur regular partitions for an odd integer $p\geq 3$. One forms a subset of the set of $p$-strict partitions, and the other forms that…
This work follows the spirit of Andrews' series of papers on Partition Analysis. In $2011$, Savage and Sills found new sum sides for the little G\"ollnitz identities and provided their partition interpretations. It turns out that similar…
Inspired by Andrews' 2-colored generalized Frobenius partitions, we consider certain weighted 7-colored partition functions and establish some interesting Ramanujan-type identities and congruences. Moreover, we provide combinatorial…
A generalization of a beautiful $q$-series identity found in the unorganized portion of Ramanujan's second and third notebooks is obtained. As a consequence, we derive a new three-parameter identity which is a rich source of…
The Rogers-Ramanujan identities are investigated using the Cauchy identity for Schur functions.
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…
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…
The Euler product for the Landau--Ramanujan constant could have motivated a curious identity by Ramanujan that appears in his notebooks two times. This observation involves a square root and the first four prime numbers of the form $4n+3$,…
The theory of Bailey's transform provides a systematic method for deriving $q$-identities, the key factor of which is the Bailey pair. The concept of Bailey pair was first extended to bilateral version by Paule. In this paper, following…
Ramanujan recorded four reciprocity formulas for the Roger-Ramanujan continued fraction. Two reciprocity formulas each are also associated with the Ramanujan--G\"ollnitz--Gordon continued fraction and a level-13 analog of the…
Andrews and El Bachraoui recently studied integer partitions where the smallest part is repeated a specified number of times and any other parts are distinct. Their results included two ``surprising identities'' for which they requested…
We construct Andrews-Gordon type evidently positive series as generating functions of partitions satisfying certain difference conditions in six conjectures by Kanade and Russell. We construct generating functions for missing partition…
In a previous paper, the author gave a combinatorial proof and refinement of Siladi\'c's theorem, a Rogers-Ramanujan type partition identity arising from the study of Lie algebras. Here we use the basic idea of the method of weighted words…
Recently, Andrews and Dastidar introduced the partition function $SOME(n)$, defined as the sum of all the odd parts in the partitions of $n$ minus the sum of all the even parts in the partitions of $n$. They derived its generating function…
We give a proof of a recent combinatorial conjecture due to the first author, which was discovered in the framework of commutative algebra. This result gives rise to new companions to the famous Andrews-Gordon identities. Our tools involve…
Recently, Hirschhorn and Sellers defined the partition function $a_r(n)$, which counts the number of partitions of $n$ wherein even parts come in only one color, while the odd parts may appear in one of $r$-colors for fixed $r\ge1$. The aim…
Recently, Andrews considered the partitions with parts separated by parity, in which parts of a given parity are all smaller than those of the other parity. Inspired from the partitions with parts separated by parity, we investigate the…
As the $q$-analog of Chebyshev polynomials, $q$-Hermite polynomials form a cornerstone in the family of $q$-orthogonal polynomials, which play a fundamental role in quantum algebra and mathematical physics. Recently, Andrews obtained a…