6 problems

Let $k,s$ be integers with $1\leqslant s\leqslant k+1$, let $a\in\{-1,1\}$, and let $\sigma_1,\dots,\sigma_k$ be fixed as in the source's definition. With $(q)_n$, Gaussian $q$-bin…

Let $\operatorname{GK}_{c}(z,q)$ denote the two-variable generating function for cylindric partitions, and let $(q)_n$ and $\genfrac{[}{]}{0pt}{}{a}{b}$ denote the $q$-Pochhammer s…

Let $k,s$ be integers with $k\geqslant2$ and $1\leqslant s\leqslant k$, and let $(q)_n$, Gaussian $q$-binomial coefficients, and $\theta$ have their standard meanings. The…

Let $k$ and $s$ be integers satisfying $1\leqslant s\leqslant k+1$, and let $(q)_n$ denote the $q$-Pochhammer symbol, $\genfrac{[}{]}{0pt}{}{a}{b}$ the Gaussian $q$-binomial coeffi…

Let $r\geq 2$ and $1\leq i\leq r$. Let $\mathcal{B}_{r,i}$ be the set of partitions whose $(i-1)$-bottom dissection consists of $i-1$ successive bottom squares followed by bottom r…

Let $r\geq 2$ and $1\leq i\leq r$. Let $\mathcal{C}_{r,i}$ be the set of partitions $\lambda=(\lambda_1,\dots,\lambda_s)$ with at most $i-1$ parts equal to $1$ and satisfying eithe…