2 problems

Let $\mathsf{Cat}_{a,b}(q,t)$ be the rational $q,t$-Catalan number and let $[m]_q$ and $\genfrac{[}{]}{0pt}{}{a+b}{a,b}_q$ denote the $q$-integer and $q$-binomial coefficient. Rati…

Fix $a,b\in\mathbb{N}$ with $\gcd(a,b)=1$. Let $\mathrm{D}(a,b)$ be the set of partitions whose diagrams fit in the triangle with vertices $(0,0)$, $(0,a)$, and $(b,a)$. For a cell…