On super Catalan polynomials

We present a $q$-analog of the super Catalan number $(2m)!(2n)!\mathrm{/}2m!n!(m+n)!$, which also generalizes the $q$-Catalan numbers $c_n(\lambda )$, due to F"urlinger and Hofbauer, for $\lambda =0$ and $\lambda =1$. We give a combinatorial interpretation for this analog when $m=2$.