Let $[n]=\{1,2,\dots,n\}$ and fix $r\in[n]$. Define the set $\mathcal{B}_{n,r}$ as the set of all set partitions of $[n]$ into disjoint non-empty blocks such that each block $B$ satisfies:
$\,\,\,\,\,\,\,\,\,\,$ either $\vert B\vert=1$, or $r\in B$, or there exist $i,j\in B$ such that $i<r<j$.
For example, there are $5$ set partitions of $\{1,2,3\}$, namely $$\{1,2,3\}; \{1\},\{2,3\}; \{2\},\{1,3\}; \{3\},\{1,2\}; \{1\}, \{2\},\{3\}.$$ If $r=1$, then we have $\vert\mathcal{B}_{3,1}\vert=4$ since $$\mathcal{B}_{3,1}=\{\{\{1,2,3\}\}, \{\{2\},\{1,3\}\}, \{\{3\},\{1,2\}\} \{\{1\},\{2\},\{3\}\}\}$$
Question. Is there any study of these sets $\mathcal{B}_{n,r}$ or their enumeration $\vert\mathcal{B}_{n,r}\vert$, directly or equivalently?