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We generalize the theory of linked partition ideals due to Andrews using finite automata in formal language theory and apply it to prove three Rogers--Ramanujan type identities of modulo 14 that were posed by Nandi through vertex operator…

In the first three papers, we conducted a series of discussions on the statistics of strict partitions and Rogers-Ramanujan partitions, specifically the sequences of odd length (denoted as $\mathrm{sol}$) and its extensions. We established…

By studying non-commutative series in an infinite alphabet we introduce shift-plethystic trees and a class of integer compositions as new combinatorial models for the Rogers-Ramanujan identities. We prove that the language associated to…

In this survey article, we present an expanded version of Lucy Slater's famous list of identities of the Rogers-Ramanujan type, including identities of similar type, which were discovered after the publication of Slater's papers, and older…

We utilize the technique of staircases and jagged partitions to provide analytic sum-sides to some old and new partition identities of Rogers-Ramanujan type. Firstly, we conjecture a class of new partition identities related to the…

We construct a family of partition identities which contain the following identities: Rogers-Ramanujan-Gordon identities, Bressoud's even moduli generalization of them, and their counterparts for overpartitions due to Lovejoy et al. and…

Machine learning components commonly appear in larger decision-making pipelines; however, the model training process typically focuses only on a loss that measures accuracy between predicted values and ground truth values. Decision-focused…

In this paper, we present a new Rogers--Ramanujan type identity for overpartitions by extending the asymmetrical version of Schur's theorem due to Lovejoy to a broader class of infinite products. More precisely, we provide a combinatorial…

Kanade and Russell conjectured several Rogers-Ramanujan-type partition identities, some of which are related to level $2$ characters of the affine Lie algebra $A_9^{(2)}$. Many of these conjectures have been proved by Bringmann,…

We prove an infinite family of linear identities for the number $A_5(n)$ of partition pairs of $n$ with $5$-cores by using certain theta function identities involving the Ramanujan's parameter $k(q)$ due to Cooper, and Lee and Park.…

We propose a method to construct a variety of partition identities at once. The main application is an all-moduli generalization of some of Andrews' results in [5]. The novelty is that the method constructs solutions to functional equations…

We explore some connections between vectors of integers and integer partitions seen as bi-infinite words. This methodology enables us to give a combinatorial interpretation of the Macdonald identities for affine root systems of the seven…

A partition of degree $n$ is a decomposition $n=i_1+i_2+\dots+i_q$, where ${i_1,i_2,\dots,i_q}$ are positive integers called the parts of the partition. Let $\lambda>0$ be an integer. The partition is said to be a $\lambda$--partition if…

We give an easy approach to L. Slater's Bailey pairs A(1)-A(8) with the help of q-Lucas polynomials.

In our previous paper, we determined a unified combinatorial framework to look at a large number of colored partition identities, and studied the five identities corresponding to the exceptional modular equations of prime degree of the…

We use an integral method to establish a number of Rogers-Ramanujan type identities involving double and triple sums. The key step for proving such identities is to find some infinite products whose integrals over suitable contours are…

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…

In 2010, Andrews considers a variety of parity questions connected to classical partition identities of Euler, Rogers, Ramanujan and Gordon. As a large part in his paper, Andrews considered the partitions by restricting the parity of…

The Rogers-Ramanujan-Gordon identities generalize the classical partition identities discovered independently by L. J. Rogers and S. Ramanujan. In 2021, Afsharijoo provided a commutative algebra proof of the Rogers-Ramanujan-Gordon…

We give manifestly positive Andrews-Gordon type series for the level 3 standard modules of the affine Lie algebra of type $A^{(1)}_2$. We also give corresponding bipartition identities, which have representation theoretic interpretations…