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
$$\mathcal{B}\_{3,1}=\{\{\{1,2,3\}\}, \{\{2\},\{1,3\}\}, \{\{3\},\{1,2\}\} \{\{1\},\{2\},\{3\}\}\}$$



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> **Question.** Is there any study of these sets $\mathcal{B}\_{n,r}$ or their enumeration $\vert\mathcal{B}\_{n,r}\vert$, directly or equivalently?
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Is this variant on set partition explored?
Here is an alternative approach to what is proposed by Brendan. 


The key idea is to notice that the only forbidden parts in the set partitions are non-singleton subsets of $\{1,2,\dots,r-1\}$ and $\{r+1,r+2,\dots,n\}$, which enables one to employ the inclusion-exclusion principle to enumerate the set partitions without such forbidden parts.


First, we consider the associated Stirling numbers of second kind ([A008299](http://oeis.org/A008299)), which enumerate set partitions without singleton parts:
$$S'(n,k) = \sum\_{j=0}^k (-1)^j\cdot \binom{n}{j}\cdot S(n-j,k-j),$$
where $S(\cdot,\cdot)$ are the conventional Stirling numbers of second kind.


Now, we can claim that
$$\begin{split}|\mathcal B\_{n,r}| &= \sum\_{p=0}^{r-1} \sum\_{q=0}^{n-r} B\_{n-p-q}\cdot \binom{r-1}{p}\cdot \binom{n-r}{q}\cdot \sum\_{i=0}^{\lfloor p/2\rfloor} \sum\_{j=0}^{\lfloor q/2\rfloor} (-1)^{i+j}\cdot S'(p,i)\cdot S'(q,j)\\
&=\sum\_{p=0}^{r-1} \sum\_{q=0}^{n-r} B\_{n-p-q}\cdot \binom{r-1}{p}\cdot \binom{n-r}{q}\cdot {\bar B}\_{p+1}\cdot {\bar B}\_{q+1},
\end{split}$$
where $B\_{m}$ are Bell numbers ([A000110](https://oeis.org/A000110)), ${\bar B}\_{m}$ are complementary Bell numbers ([A000587](https://oeis.org/A000587)), $i$ and $j$ stand for the number of forbidden parts from $\{1,2,\dots,r-1\}$ and $\{r+1,r+2,\dots,n\}$ respectively, and $p$ and $q$ stand for the size of the unions of these forbidden parts.


