# Counting Set Partitions With A "Central" Condition

Let $[n]$ denote the set $\{1,2,\cdots,n\}$. A partition of $[n]$ is defined in the usual way to be a collection $(B_1,\cdots,B_k)$ of disjoint non-empty subsets (blocks) such that $[n] = B_1 \cup B_2 \cup \cdots \cup B_k$. A partition is called $m$-regular if $$i_1, i_2 \in B_j \implies |i_1-i_2| \geq m.$$

Let $p(n,k)$ denote the number of partitions of $[n]$ into $k$ blocks and $p(n,k,m)$ the number of $m$-regular partitions of $[n]$ into $k$ blocks. We know that $p(n,k)$ is the Stirling number of the second kind and one can show that for $m \geq 2$ we have $p(n,k,m)=p(n-1,k-1,m-1)$.

Call a partition $r$-central (I have no idea if this concept already exists under another name) for some $1 \leq r \leq n$ if every block satisfies either $$\bullet \quad |B_j| = 1,$$ $$\bullet \quad r \in B_j, \quad \text{or}$$ $$\bullet \quad \exists i_1,i_2 \in B_j\quad \text{ such that } \quad i_1 < r \, \wedge i_2 > r .$$

In words we require that all non-singleton blocks contain at least one element from either side of the point $r$ (inclusive). Is there an expression for the number of $r$-central partitions $p_r(n,k)$? What about $r$-central and $m$-regular?

Well i used them to count some generalizations of undecomposable permutations so i called them $r$-idecomposable partitions.
In your notation i obtained: $$p_r(n,k)=\sum _{a=0}^{r-1}\sum _{b=0}^{n-r}\sum _{i=0}^{r-a-1} \sum _{j=0}^{n-r-b}\underbrace{\binom{r-1}{a}\binom{n-r}{b}}_{\text{elements to be in block containing r}}\underbrace{\binom{r-a-1}{i}\binom{n-r-b}{j}}_{\text{singleton blocks}}\\ \underbrace{ {r-a-1-i\brace k-1-i-j}{n-r-b-j \brace k-1-i-j}(k-1-i-j)!}_{\text{a matching in between blocks on the left and blocks on the right}}.$$ There are a lot of cools things here.