An overpartition analogue of the $q$-binomial coefficients

TL;DR

This paper introduces overpartition analogues of $q$-binomial coefficients, explores their properties, and applies them to prove a Rogers-Ramanujan type partition theorem, expanding the combinatorial understanding of overpartitions.

Contribution

It defines over Gaussian polynomials as overpartition analogues of $q$-binomial coefficients and investigates their properties and applications.

Findings

Defined over Gaussian polynomials as overpartition analogues

Proved recurrences and combinatorial interpretations

Established a Rogers-Ramanujan type partition theorem

Abstract

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 Gaussian polynomials or over $q$-binomial coefficients. We investigate basic properties and applications of over $q$-binomial coefficients. In particular, via the recurrences and combinatorial interpretations of over q-binomial coefficients, we prove a Rogers-Ramaujan type partition theorem.