  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.
