2 problems

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…