The Gaussian coefficient revisited

TL;DR

The paper introduces a new, more compact $q$-$(1+q)$-analogue of the Gaussian coefficient, based on Boolean algebra decompositions, extending to posets and discussing higher analogues.

Contribution

It presents a novel, simplified $q$-$(1+q)$-binomial and extends the concept to Birkhoff transforms and higher analogues, improving upon previous formulations.

Findings

New $q$-$(1+q)$-binomial is more compact than previous versions.

Underlying structure involves Boolean algebra decomposition of posets.

Extensions to Birkhoff transforms and higher analogues are discussed.

Abstract

We give a new $q$-$(1+q)$-analogue of the Gaussian coefficient, also known as the $q$-binomial which, like the original $q$-binomial $\genfrac{[}{]}{0pt}{}{n}{k}_{q}$, is symmetric in $k$ and $n-k$. We show this $q$-$(1+q)$-binomial is more compact than the one discovered by Fu, Reiner, Stanton and Thiem. Underlying our $q$-$(1+q)$-analogue is a Boolean algebra decomposition of an associated poset. These ideas are extended to the Birkhoff transform of any finite poset. We end with a discussion of higher analogues of the $q$-binomial.