Does this discrete distribution have a name/literature? In my work on discrete urn problems, I have run across a family of discrete distributions with an unusual form, and I am trying to find out if this family of distributions is in the statistical literature.  To do this, I would like to know if it has a name.  Let $S(n,k,r)$ denote the non-central Stirling numbers of the second kind, and let $S(n,k)$ denote the (central) Stirling numbers of the second kind.  It is well-known that these two types of numbers are related by:
$$S(n,k,r) = \sum_{t=k}^n {n \choose t} r^{n-t} S(t,k).$$
Thus, we can form a valid family of discrete distributions with probability mass function:
$$p(t|n,k,r) = {n \choose t} \frac{r^{n-t} \cdot S(t,k)}{S(n,k,r)}
\quad \quad \quad \text{for } t = k,...,n.$$
I have encountered this distribution in some analysis I am doing on urn problems, and I have managed to derive some of its properties.  I am trying to identify if this distribution has been examined in the academic literature, which is hard to do, since I don't know its name.

My Question: Does this distributional family have a name?  Is there any existing academic literature on this distribution that anyone can point me to?
 A: 1. Relation with minimal covers
This answer is not a distribution but a related set of numbers.
For the cases with $r = 2^{k}-k-1$ you can express the probability $p(t \vert n,k,r)$ by using numbers relating to minimal covers of a set with cardinality $n$ ( from 'Minimal covers of finite sets' by T.Hearne and C.Wagner):
$$p(t \vert n,k,r=2^{k}-k-1) = \frac{M(n,k,t)}{M^\star(n,k)}$$

*

*Here $M(n,k,j)$ is the number of $k$ element minimal covers of a $n$-set, such that $j$ points are uniquely covered (where $j\geq k$ since each element of a minimal cover must contain at least one unique member).
$$M(n,k,j) = {{n}\choose{j}}\left( 2^{k}-k-1\right)^{n-j}S(j,k)$$

*And $M^\star(n,k)$ is the number of $k$ element minimal covers of a $n$-set.
$$M^\star(n,k) = \sum_{j=k}^n M(n,k,j)$$
2. Relation conditional urn and balls problem
This answer is not a solution of a related distribution but it is how I imagined the problem could be when I was trying to reverse engineer what kind of urn problem is underlying the formula. I believe it may be helpful for others to visualize the problem.
This is how I got to think of the related urn problem: First consider that the expression ${{n}\choose{t}}S(t,k)$ is the number of ways to select $t$ balls from a total of $n$ balls and put them into $k$ identical urns such that each urn is non-empty (this term also occurs also in the referenced article of Hearne and Wagner). Then, the difference with the expression in the question is the term $r^{n-t}$, and this resembles the number of ways to distribute the remaining $n-t$ balls among $r$ urns. So this leads to the following urn-problem.
Put $n$ balls into $k+r$ urns. Let $T$ be the number of balls in the first $k$ urns, let $E$ be the event that each of these $k$ urns is non-empty. Then $$P(T=t \vert E) = \frac{ {{n}\choose{t}}r^{n-t}S(t,k) }{\sum_{j=k}^n{{n}\choose{j}}r^{n-j}S(j,k)} $$
